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# Worked example: Derivatives of sin(x) and cos(x)

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.A (LO)
,
FUN‑3.A.4 (EK)

## Video transcript

what we want to do is find the derivative of this G of X and at first it could look intimidating we have a sine of X here we have a cosine of X we have this crazy expression here with a PI over cube root of x we're squaring the whole thing at first it might seem intimidating but as we'll see in this video we can actually do this with the tools already in our toolkit using our existing derivative properties using what we know about the power rule which tells us the derivative with respect to X of X to the N is equal to n times X to the N minus 1 we've seen that multiple times we also need to use the fact that the derivative of cosine of X is equal to negative sine of X and the other way around the derivative with respect to X of sine of X is equal to positive cosine of X so using just that we can actually evaluate this or evaluate G prime of X so pause the video and see if you can do it so probably the most intimidating part of this because we know the derivative of sine of X and cosine of X is this expression here and we can just rewrite this or simplify it a little bit so it takes a form that you might be a little bit more familiar with so actually let me just do this on the side here so pi PI over the cube root of x squared well that's the same thing this is equal to PI squared over the cube root of x squared this is just exponent properties that we're dealing with and so this is the same thing we're going to take X to the 1/3 power and then raise that to the second power so this is equal to PI squared over let me write it this way I'm not going to skip any steps because this is good review of exponent property is X to the one-third squared which is the same thing as PI squared over X to the two-thirds power which is the same thing as pi squared times X to the negative two-thirds power so when you write it like this it starts to get into a form you're like oh I can see how the power rule could apply there so this thing is just pi squared times X the negative 2/3 power so actually let me delete this so this thing can be rewritten this thing can be rewritten as pi squared times X to the negative to the negative 2/3 power so now let's take the derivative of each of each of these each of these pieces of this expression so we're going to take where you want to evaluate what G prime of X is so G prime of X is going to be equal to you could view it as the derivative with respect to X of 7 sine of X so we could take let's do the derivative operator on both sides here just to make it clear what we're doing so we're going to apply it there we're going to apply it there and we're going to apply it there so this derivative this is the same thing as this is going to be 7 times the derivative of sine of X so this is just going to be 7 times cosine of X this one over here this is going to be 3 or we're subtracting so it's going to be this subtract this - we can bring the the constant out that we're multiplying the expression by and the derivative of cosine of X so it's minus 3 times the derivative of cosine of X is negative sine of X negative sine of X and then finally here in the yellow we just apply the power rule so we have the negative 2/3 it's actually let's not forget this minus sign I'm going to write it out here and so you have the negative 2/3 you multiply the exponent times the coefficient it might look confusing PI squared but that's just a number so it's going to be negative and then you have negative 2/3 times pi squared times pi squared times X to the negative 2/3 minus 1 power negative 2/3 minus 1 power so what is this going to be so we get G prime of X is equal to is equal to 7 cosine of X and let's see we have a negative 3 is a negative sign of X so that's a positive three sine of X and then we have we're subtracting but then this is going to be a negative so that's going to be a positive so we could say plus 2 PI squared over 3 2 pi squared over 3 that's that part there times X to the so negative 2/3 minus 1 we could say negative 1 and 2/3 or we could say negative 5/3 power negative 5/3 power and there you have it we were able to tackle this thing that looked a little bit hairy but all we get to use was the power rule and what we knew to be the derivatives of sine and cosine
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