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Problem 41 Hard Difficulty

Find the derivative of the function.
$ f(t) = \sin^2 (e^{\sin^2 t}) $


$=4 \sin \left(e^{\sin ^{2} t}\right) \cos \left(e^{\sin ^{2} t}\right) e^{\sin ^{2} t} \sin t \cos t$

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Video Transcript

All right. So we're going to use the chain rule to find the derivative of this function, and this is going to test our chain rule to the limits here. First, I'm going to rewrite the function. So when you see sine squared the square dis really on the very most outside of it. So let's rewrite this as the sign of E to the it's get that e a little bit better e to the power of sine of t squared. So I moved that squared symbol as well, and then I'm going to move the other squared symbol to the very outside. So let's check this out. We have many, many layers to go here. Okay, so we start finding are derivative by taking the derivative of the outermost function, the squaring function. So we bring down the to and we raised everything inside it to the power one. Let's see if I have enough parentheses there of it. Now we move on to the next layer, which is sign, and we take its derivative, which is co sign so coastline of everything inside it there didn't need as many parentheses that time. Now we move on to the next layer, which is e to a power. And we do that as each of that power times the derivative of the power. So now, for the derivative of the power that has two layers, we have the squared on the outside. So we bring down the two and raise signed T to the first. And then we do the derivative of sign, which is co sign. Now we take a look and we ask ourselves, Is there anything we can do to make this simpler? Do we have any factors that are the same as other factors that we could combine? And the answer is no. But what we could do is take the two times the two and make that a four. And then any time we had a square that wasn't written in its proper position, we could put it back into its proper position. So we have four times the sign of e to the sine squared t times a co sign, uh, v to the signs where t times e to the sine squared T time scientist E times ko 70. Now you could have these terms in a variety of orders. It won't matter, like if you look up the answer in the book and they have all the same factors, but they're in a different order. It's okay. If they're factors they could be rearranged.