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 = tan 5x²

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 is equal to G of X and the outer function Y is equal to f of U, or the composition Y is equal to F of G of X. And then we need to find the derivative of Y with respect to X. We're given the function Y is equal to the tangent of 8X4. We're given 4 possible choices as our answers. For choice A, we have U is equal to 8 x 4, F of U is equal to 2 of view, and the derivative of Y with respect to X is equal to 32 x cubed multiplied by c2 of 8X4. For choice B, we have U is equal to 8 X. F of U is equal to 2 of U, and the derivative of Y with respect to X is equal to 32 x cubed multiplied by c2 of 8X4. For choice C we have U is equal to 8 x 4, F of U is equal to tangent of u, and the derivative of Y with respect to X is equal to 32X4 multiplied by c2 of 8X4. And for choice d, we have U is equal to 8 x 4th, F of U is equal to 10, and the derivative of Y with respect to X is equal to 32 x cubed multiplied by squad of 8X4. Now, the first thing we need to do for this question is to identify the inner and outer functions. Now, our inner function needs to be a function of X. So for our inner function, we're going to choose the argument of the tangent function of our function Y. So that means that you is going to be equal to 8 X the 4th, and so that is going to be our inner function. So now that we've identified our inner function, We can write why. As a function of you, and so this is going to be our outer function. And so if we rewrite Y, replacing ADX to the 4th with you, this is going to be equal to tangent of you. And so that is going to be our outer function. So now that we have the inner and outer functions identified, we can now calculate the derivative of Y with respect to X, and we're going to use this, use it doing, sorry, using the chain rule. So, recall for the chain rule that the derivative of Y with respect to X is equal to the derivative of Y with respect to U. Multiplied by the derivative of U with respect to X. And so we're going to use our inner and outer functions to calculate these derivatives. So, the first derivative that we're going to calculate is the derivative of U with respect to X, and this is going to be equal to the derivative with respect to X of the quantity of 8 X to the 4th. So here we need to recall our power rule for derivatives, and recall for that that we have the derivative with respect to X of the quantity of X-rays to the N. And this is going to be equal to N multiplied by X raised to the N minus 1. So, if we apply the power rule to our derivative of U with respect to X, this is going to be equal to 32. Execute We also need to calculate our derivative of Y with respect to you. And this is going to be equal to the derivative of with respect to you, of the tangent of you. And recalled that the derivative with respect to X, Of tension of X. is equal to Deacon squared X So applying that rule, where the derivative tangent, or derivative of Y with respect to you is going to be equal to the squared of you. So, now we have calculated both these derivatives, we can substitute their results back into our chain rule formula. We'll have the derivative of Y with respect to X is equal to the derivative of Y with respect to U, which we calculated to be squared of you. However, we want our final answer as a function of X. So for you, we're going to replace it with 8 X to the 4th. So we're going to have squared. Of 8 X the 4th. And then this quantity is going to be multiplied by the derivative of view with respect to X. And so, we'll be multiplying our sequence squad of 8X4th by our derivative of U with respect to X, which we calculated to be 32 X cubed. So just rearranging. There are uh multiplication there, we will have 32 excubed, multiplied by Deacon squared of 8 X the 4th. And so this is the answer that we were looking for for our derivative. And so now we've identified both our inner and outer functions and calculated the derivative, we can look at our 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 = tan 5x²

Key Concepts

Composite Functions

Chain Rule

Inner and Outer Functions

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