5–24. For each of the following composite functions, find an i...

Question

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = sin⁵x

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says to identify the inner function U equal to G of X, and the outer function Y is equal to FFU, where the composition Y is equal to F of GX, and then find the derivative of Y with respect to X. And we're given the function Y is equal to cosine to the 11th of X. We give 4 possible choices as our answers. For choice A, we have U is equal to sin of x, F of U is equal to u 11th, and the derivative of Y with respect to X is equal to -11, multiplied by s10th of X, multiplied by cosine of X. For choice B, we have U is equal to sin of x, F of U is equal to u 11th, and the derivative of Y with respect to X is equal to -11, multiplied by cosine to the 10th of X multiplied by sin of x. For choice C we have U is equal to cosine of X. F of u is equal to 11th, and the derivative of Y with respect to X is equal to 11, multiplied by cosine to 10th of X, multiplied by sin of x. And for choice d, we have U is equal to cosine of X, F of u is equal to u 11th, and the derivative of Y with respect to X is equal to -11 multiplied by cosine to the 10th of X multiplied by sin of x. Now, the first part of this question asks us to identify our inner function U, which is equal to G of X. So I need you to be a function of X. So if I look at the function that we're given, the Y is equal to cosine to the 11th of X, we're going to want to set u equal to. Cosine of X Then we can use that to write our outer function, Y, which is a function of U. And if we replace cosine of X with U in our function for Y that we were given, that means that our Y is going to be equal to U 11th. So the U equal to cosine of X is our inner function, and Y, which is a function of u equal to u 11, is our outer function. So that's the first part of the question answered. Now we need to find the derivative of Y with respect to X. And for this, we're going to use our chain rule. So recall for the chain rule that we have the derivative of Y with respect to X, it's going to be equal to. The derivative of y with respect to you multiplied by the derivative of u with respect to x. So, now that we've identified our inner and outer functions, we can use those to calculate these derivatives in the chain rule. So, the first thing we're going to calculate is the derivative of U with respect to X. So this is going to be equal to the derivative with respect to X. Of cosine of X. And recall that when you take the derivative of cosine of X with respect to X, that will give you minus sin of X. Now we need to take our derivative of Y with respect to you. And this is going to be equal to the derivative with respect to you. Of the quantity of you to the 11th. And this is going to be equal to 11, multiplied by U raised to the 10th. So recall that when you um take the derivative of a variable raised to a power, that you multiply that variable by the power, in this case 11, and then we need to subtract one from the exponent. So we'll have 11 minus 1, which is 10. So now we can substitute these expressions back into our formula for the chain rule. And so we'll have the derivative of Y with respect to X is equal to our derivative of Y with respect to U, and so that is going to be 11. Multiplied by U rate of the 10th, but we want our final answer to be a function of X. So for you, we're gonna substitute back in cosine of X. So, this is gonna be 11, multiplied by Cosine to the 10th of X. And so this quantity is now going to be multiplied by. A quantity of the derivative of U with respect to X. And we calculated that, and that is equal to minus. Sign of Xs. And so we can just simplify this expression, so this is going to be equal to -11, multiplied by cosine to the 10th of X, multiplied by sin of X. And so that is the answer for a derivative that we were looking for. And so now that I have all my um questions answered, I can look at my answer choices, and the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = sin⁵x

Key Concepts

Composite Functions

Chain Rule

Power Rule

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