Calculus 1 - Derivatives

In this lesson we will focus on finding the derivative of a function, so let's start with a constant. The derivative of any constant is equal to zero. For example, the derivative of the constant 5 is 0 and the derivative of, say, negative 7. is also 0 and you may be wondering what exactly a derivative is. A derivative is a function that gives you the slope at some value of x. So let's say that if we have the function f of X is equal to 8, if we graph this function. It would look like a straight line at y is equal to 8, so around this region let's say it's a ye Queene.

Now, what is the slope of a straight line? The slope of a straight line is 0, so if you found the derivative of this function represented. for F prime of X that will give you 0 if you see D over DX it means you are about to differentiate something with respect to find the derivative of a monomial, for example, what is the derivative of x squared? Now there is something called the power rule and the power rule is very useful for finding the derivative of monomials, so here is the formula you want to use for the derivative of a high variable. to a constant like derivative of x squared will be 2 X to the power of 2 minus 1, which is 1 or basically 2 to the power of X, so that's the derivative of x squared.

Now let's try other examples, using the same formula which is the power rule, go ahead and find the derivative of these functions, so find the derivative of X cubed X to the fourth power and also X to the fifth power, so the derivative of X cubed will be 3 in this case Anna Story, so it's 3 is 4 so it will be 4 X to the power of 4 minus 1 and 4 minus 1 is 3 so it is 4 X To the fourth power and that is simple way in which you could find the derivative of a function now say that if you want to find the 4X derivative to the seventh power, how would you?

How would you find the derivative of that particular monomial? So what we have to do is use something called the constant multiple rule and here is the derivative of a constant multiplied by a function with, say, a monomial, it will be the constant multiplied by the derivative of that monomial or that function, let's just put f of X here, so in this case our C value is 4 and f of Let's get the 4 out front and then we're going to multiply it by the derivative of x to the seventh. Now, using the power rule, we can find the derivative of that function, so it's going to be 7.

X to the power of 7 minus 1, so 7 minus 1 is 6, so we have. 7 monomials 8 to 4 so the derivative of x to 4 will be 4 times for the next one let's move the constant to the front and then we're going to multiply by the derivative of X to the sixth power so we can take this exponent and move it to the front so this is going to be 5 times the derivative of X to the power of 6 minus 1 which is 5 six let's prove the derivative of 9 of X to the fifth power is 5 X to the fourth power and then 9 times 5 is going to be 45 so the answer is 45 the seventh power and the derivative of power now we said that the derivative of x squared is equal to 2x now how do we know that by the way let's say if f of X is x squared, that means that the derivative of f of X, which is F prime of but how can we confirm this now?

Remember that the derivative is a function that can give you the slope at some value of x, so we're going to show that soon, but first, is there another way we can get this answer besides using the power rule? In a typical

calculus

course you need to know what that form is and it is sometimes known as the definition of the derivative, maybe you can I have seen this function f prime of X is equal to the limit as H approaches zero of f of X plus h minus f of , what is f of X plus h, all you need to do is replace X with X plus h, so in here we had x squared, so instead of plus h is X plus h squared.

Now we're going to plug it into this formula, so let me delete this, so remember we're trying to prove that F prime of X is equal to 2x. I'm going to delete that soon now we have this F prime of and now that? Do you think we should do at this point what is our next step? Our next step is to frustrate that expression, so this is equivalent to the limit as H approaches 0 and X plus h squared is the same as X plus h times X plus h, so. let's go ahead and frustrate that expression now when you take a

calculus

exam you will need to rewrite the limiting expression even though it may be tedious some teachers will take points off you if you don't rewrite it so here we have x multiplied by squared and then we have x times H and then it's H times X, which is the same as x times H and then the last H times H, so that's H squared and then minus x squared divided by H now, in At this point, we can cancel the x squared term and we can combine like terms XH plus XH, that's 2 greatest common divisor, which is H, so if we take an H out of 2 H approaches 0 from 2 of x squared is 2x and this is how you can find this answer using the limit process now we said that the derivative is a function, we will give it the slope at any value of x, so let's say that f of X is x squared and we want to find the slope of the tangent line at x is equal to 1, so knowing what is F prime of of X when it would look like this and when This is what the derivative function tells you, it gives you the slope of the tangent line at some output value, now you need to know the difference. between a tangent line and a secant line a secant line is basically a line that touches the curve at two points and I missed it so let's do it again and the tangent line is a line that touches the curve only at one point so make sure Now you know the difference between the two, now in algebra you have learned that to find the slope of a line you need two points and this is basically finding the slope of a secant line that is on a curve, so let's put M secant and that's it you know that since y2 minus y1 is equal to x2 minus x1 so we could take two points on this curve and basically get a secant line and when those two points approach each other at this point the slope of the secant line approaches the slope of the line tangent now we need to choose two points where the midpoint of those two points is x is equal to 1, so we can choose, say, 0.9 and 1.1 as our values ​​of x1 and x2 because if you add those two numbers and You divide them by 2, the average of 0.9 and 1.1 is 1 or we could choose the point at the high point 9 and 1 point 0 1 because the midpoint of those two numbers is still 1, however 0.99 and 1 point 0 1 is closer to 1 than point 9 and 1 point 1, so the slope of the secant line based on these two values ​​will be much closer to the slope of the tangent line at x is equal to 1 so let's go ahead and calculate those values ​​so let's say x1 is 0.9 to start and x2 is 1 point 1 and let's use this formula to calculate the slope of the secant line now note that the slope of the line tangent is this number, it is equal to 2, so it will be Y to the power of negative y1 divided by x2 minus x1, so y2 corresponds to the value of Y for this value of x. and Y is equal to f of X, so we could use this function f of x is equal to x squared to find y2, so when squared when eight one divided by the point two already gives us an exact answer which is two and then there is It is not necessary to use the point nine nine in this case.

We can see that it is exactly the same, so let's try an example where it may not be exactly the same. This time let's say f of X is x cubed and we want to calculate the slope. of this tangent line at x is equal to two, then we know which F prime of times 2 squared 2 squared is 2 times 2, which is 4 times 3, is 12, so the slope of the tangent line at x is equal to 2 is 12. Now let's see if we can approximate this value with the slope of the secant line, so let's choose a value of X 1 of 1.9 and a value of that X 2 is 2.1 now what is y2 y2 has to be 2 point 1 raised to the third power because Y is equal to to the third power that will be nine point two six one and one point nine raised to the third power that is going to be six point eight five nine and two point one minus one point nine that is point two nine point two six one minus six point eight five nine that's two point four zero two and if we divide that by point two it gives us a very good approximation actually twelve point zero one and then you can see that you can approximate the slope of this tangent line using the slope of the secant line and that is what the derivative tells you, it gives you the slope of the tangent line that touches the curve at one point, that is an interview with the value of x, remember that the derivative is a function and that helps you find the slope of a tangent line at some value of x, so keep that in mind now let's talk about how to find the derivative of a polynomial function, so let's say f of of that function then what is f prime of What about the derivative of 7x squared using a constant multiple rule?

It will be seven times the derivative of x squared, which is 2x or 2 X to the first power and seven times 2x is 14x now What about the derivative of negative 8x? What does that equate to? Note that this is negative 8 times X to the first power, so it will be negative 8 times the derivative of x to the first power. Now, what is the derivative of X? to the first power, using the power rule, you need to move the 1 to the front so that it is 1 times X to the power of 1 minus 1, which becomes negative 8 times 1, the output is 0 now what is , is the output 0 or anything raised? raised to 0 is 1,so this becomes negative 8 times 1, which is just negative 8, so the derivative of negative 8x is just negative 8 and the derivative of any constant is 0, so we could stop it here.

This is the answer f prime of from patreon like you I can see the link on the screen and on that page I have other video content that might interest you, so feel free to check it out when you get a chance. Now let's go back to the lesson. Now let's say f of X is 4 X to the fifth power plus 3x to the fourth power plus 9 constant multiple we are going to rewrite the constant and take the derivative of X to the fifth power using the power rule and that will be 5x to the fourth power now the derivative of is always just 1 and the derivative of a constant is zero and then we have 4 times 5 which is 20 and 3 times 4 which is 12 and then this is the final answer f prime of X is 20 X to the fourth power plus 12x cubed plus 9 now let's say that f of 2.

Go ahead and try this problem every time we need to find a slope with the tangent line. First you need to find the derivative of the function that is. f prime of X and then just plug the value of x into that function, so let's determine F prime of X versus the derivative of the derivative of x squared is 2x and for the constant we just don't need to worry about that, so f prime of X will be 10 2 times 5 is 10 so 10 this is what we have now to calculate the slope and let's replace 2 is 12, so now we have is the slope of the tangent. line when X is equal to two now let's say f of X is 1 over You have to rewrite the function and what we have to do is take the variable Now at this point you can use the power rule, so remember the derivative of what we have minus 1 divided by x squared and this is how you can find the derivative of a rational function.

Say f of X is 1 over x squared, go ahead and find f prime of x for practice, so take a minute and try that example like before. We're going to rewrite the function so let's move the variable say of the constant, but the exponent move it to the front, so our value n is negative two and then let's subtract negative 2 times one minus two minus one is negative three and now let's rewrite this expression taking the variable you ask why I divided by one is the same as negative two X to negative 3 is the same as negative two X to negative three over one.

I just like to write it in a fraction so you can see what I'm going to do next and that's move the X variable down so now the exponent will change from negative three to positive three and this is the final answer so this is the derivative of 1 over x squared. Now let's try another example, let's say f of X, let's try a harder example, let's say it's 8 over and this will be 8 X raised to the negative fourth power and now let's differentiate. the function, so we need to use the constant multiple rule in this case, so it's going to be eight times the derivative, so I'm going to write that multiplied by D over DX, the derivative of x raised to the negative fourth power, and so on this case. using the power rule of ours and the value is negative 4, so it will be multiplied by negative 4x to the power of negative 4 minus 1, so negative 4 minus 1 is negative 5 and 8 times negative 4 is negative 32, so that the answer is negative 32 to the negative 5th power, well, let's rewrite it, although we don't leave it at that, so if we move the Now let's talk about finding the derivative of radical functions, for example, what is the derivative of the square root of x, so what do you think we should do now?

The first thing we must do is rewrite this expression as a rational exponent so that X is equal to X raised to the first power. and if you don't see an index number, it's always 2, so this is equivalent to X to the power of 1/2. Now in this form, we can use the power rule, so n will be 1/2, so it is 1/2. Now negative 1 over 2 not because we have a negative exponent we need to rewrite this so now the 1 is on top and the becomes 1 over 2 times X to the positive 1/2 and at this point I can rewrite the rational exponent as a radical so that we know that 2 square root Now let's work on another example, let's say f of X is the cube root of x to the fifth power, so what is f prime of?

X, go ahead and try that, so let's start by rewriting this expression so that the cube root of numerator of the rational exponent and the index number becomes the denominator of the fraction we see here. Now let's use the power rule, so in this case n is a fraction, it will be 5 over 3 and then we will have X raised. to N minus 1 so that's 5 over 3 minus 1 now as before we need to get common denominators 1 is the same as 3 divided by 3 and then we have 5 over 3 the minus 3 over 3 5 minus 3 is 2 so it becomes 2 over 3 and I'm going to rewrite this as 5 times X to the 2/3 divided by 3 now, because the exponent is still positive, we don't need to move the variable convert this back into a radical expression, so it will be 5 times the cube root of x squared over 3 and this is the final answer.

Here's another one you can work on, so let's say we have the monomial or rather just a radical expression the seventh root of x to the fourth power what is the derivative of that expression so this is X to the power of 4 over 7 and let's use the power rule so that n is 4 over 7 and it's going to be Negative 7 so I'm going to rewrite this as a fraction the four on top of the The variable X is currently at the top, but the 7 is at the bottom of the fraction. Now in this case, we have a negative exponent, so we need to move the variable , so the final answer will be 4/7 times the 7th root of x cubed and that's it, this is the answer now, of course, if you want, you can rationalize. the denominator but I'm not going to worry about that in this video now let's talk about other problems that you can see in your homework, so let's say if you are given a problem like this, it has x squared on the outside and then in parentheses it has X cubed plus 7 how would you find the derivative of this expression what would you do in this case the best thing you can do right now with what you already know is to distribute x squared 2x cubed plus 5 and then you can find the derivative, so x squared times X cubed will be the fifth is 5x to the fourth power and the derivative of x squared is 2x, so the final answer is 5x to the fourth power plus 14x and that's what you should do if you ever run into a situation like that.

Try a different example, let's say f of X is equal to 2x minus 3 raised to the second power, what would you do in this case? There's something called the chain rule that we can use here, but you haven't learned it yet, so we'll save it for another day or rather later in this video, something we can do is expand this expression, so that every time You see an exponent of 2, whatever the associated exponent is, it means that you have two of these things and you multiply them together. so this expression is equivalent to 2x minus 3 times another 2x minus 3 which doesn't look like a 3 and so what we're doing is multiplying one binomial by another binomial and then let's use the sheet method, so let's multiply the first two terms 2x by 2x 2 times 2 is 4 x times plus negative 6x is negative 12x and now we can find the first derivative, the derivative of x squared is 2x, the derivative of x is 1 and for a constant it is 0, so the final answer will be 8x minus 12 and that is the derivative of 2x and negative 3, so that's what you could do in a situation like this.

Now let's say we have a fraction X to the fifth power plus 6x to the fourth power plus 5x cubed divided by x squared in this. case, what is the derivative of f of X? Now, based on the examples above, you know that you need to simplify this before finding the derivative. So how can you simplify this expression if you are dividing a trinomial by a monomial? What you could do is divide each term by x squared separately, so let's start by dividing you need to subtract, so this is 5 minus 2 that's X cubed and that's going to be the first part, so X to the fifth power divided by x squared is It will be 6 x squared all you need what we do is subtract from the exponents 4 minus 2 is 2 and 5x cubed divided by now we can find the first derivative, so it's going to be 3x squared and the derivative of x squared is 2x and the derivative of of the

derivatives

of trigonometric functions and I'm I'll give you some that you need to know and for now write these down because we're going to use this later.

Now, the derivative of sine Next, you should know that the derivative of the secant X is secant X tangent X and the derivative of the cosecant The thing is that if you see a C in front you are going to have a negative sign like the derivative of the cosine, it was negative sine. Now consider the last two, the derivative of the tangent is secant squared and based on that, what do you think the derivative is? of the cotangent and here it is, so let's say that if you have two functions multiplied together and you want to find the derivative of that result, it will be the derivative of the first function multiplied by the second plus the first function multiplied by the derivative of the second, so let's say that If we want to determine the derivative of X by sine X,so in this case we could say that f is x squared and G is the sine squared is 2x now G will be equivalent to sine of the equation, so f prime is 2x G is sine Leave the answer like this and this is how you can use the product rule when finding the derivative of functions that multiply together.

Now let's try other examples. Try this problem. is the derivative of let's say 3x to the fourth power plus 7 times X cubed minus 5x. Now we can foil this expression because we did an example like that before, but let's use the product rule to get the answer, feel free to pause the video if you want so what I'm going to doThe first thing I do is write the formula so that the derivative of F times G be the derivative of the first part multiplied by the second plus the first part. Let me write that again multiplied by the derivative of the second, so which is F and which?

G we're going to say that F is the first part and then we're going to say that f is 3x to the fourth power plus seven so what is f prime so that the derivative of X to the fourth power is 4x cubed, but we? We're going to multiply that by three and then three times four will give us 12, so this will be 12x cubed and the derivative of seven is zero. Now G is the second part of the function, so G is X cubed minus 5x G. prime will be three x squared and the derivative of so using the formula this expression becomes equal to the one I'm going to write here this is going to be F prime, which is 12x cubed times G, which is X cubed minus 5x plus F and that's three seven times G prime 3x squared minus five and that's it, now here's a challenging problem for you. is the derivative of x cubed times the tangent x times three x squared minus nine, so this time we have three parts multiplied together, so we saw how to use the product rule when multiplying two different functions together, but what? what about three? different functions, so if the derivative of let's say a two-part function like F times G if that's F Prime and G Plus F G Prime, what would be the derivative of let's say F times G times H, so this will be what are we going to do? to differentiate the first part and then leave the second two parts the same, plus we are going to leave the first part this differentiates the second part and then we leave the third part the same and then it will be the first two parts multiplied by the derivative of the last part, so using the product rule, when you differentiate one part, the other two parts must remain the same and then you just go in order to differentiate the first part and then the second part and then the third part, so once you understand the format or the procedure to do this, you can go ahead and get the answer without actually writing what is F of G and H, so first let's find the derivative of the first part, the derivative of the cube x is 3x squared, now the other two parts are G. and H, we're just going to rewrite it for now, so it's going to be multiplied by the tangent X and then multiplied by 3x squared minus 9.

Now let's rewrite the first part, which is the cube second. part the derivative of the tangent if you remember it is secant squared now let's rewrite the third part which is 3x squared minus 9 now for the last part we are going to rewrite the first two parts X cube and tangent X but this time we are going to take the derivative of 3x squared minus 9, which will just be 6x because it's going to go to 0. Now let's say if we want to find the derivative of a fraction like let's say 5x plus 6 divided by 3x minus 7, in this case I want to use something called the quotient rule and here it is the formula that we are going to use, then the derivative of let's say F divided by G will be G F - f G prime divided by G squared and this is something that I just need to memorize, you just have to know that function, at least it works for me when I was in secondary school.

I just memorized that function now. F will be the top of this function, so in this case F will be 5x + 6 F prime is the derivative of F, so the derivative of 5x + 6 is 5 now G G will be the bottom of this function, so what G will be 3x - 7 which means the story of G Prime so if it helps write everything down by all means go ahead and do it if it makes your life easier or if it helps you avoid mistakes and in an exam one of the most important things you should do is avoid mistakes because if you make a mistake even if you already know it, I mean that will ruin your score on the exam, now let's go ahead and finish this G is 3x minus 7 and then F prime is 5 and then we have F which is 5x plus 6 and then G prime is 3 and then we divide by G squared, so G is 3x minus 7 and then we square that.

Now in this case I'm going to simplify because it doesn't require much work to do, so let's start by distributing the 5 to 3x - 7, that is, 5 times 3x. that's 15 and I'm in the hallway with the stuff below because it looks better so now we can cancel out 15 answer, feel free, but sometimes, if it takes a lot of work to simplify it, most teachers will let you write the answer as is. Some teachers will let you leave the answer like that, so basically you need to do it. know your teacher and how he wants you to write the final answer