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 = e^4x²+1

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 f of U, or the composition Y is equal to F of G of X, and then find the derivative of Y with respect to X. We're given the function Y is equal toe raised to the quantity of 5 x2 + 21. We're given 4 possible choices as our answers. For choice A, we have U is equal to 5 x2 + 21. F of U is equal to eu. And our derivative of Y with respect to X is equal to E raised to the quantity of 5 X2 + 21. For choice B, we have U is equal to 5X2 + 21, F of U is equal to E U, and the derivative of Y with respect to X is equal to 10 multiplied by E raised to the quantity of 5 X2 + 21. For choice C we have U is equal to 5x2 + 21, F of U is equal to eu, and the derivative of Y X is equal to 10x multiplied by E raised to the quantity of 5 X2 + 21. And for choice D we have U is equal to 5 x 2, F of U is equal to eu, and the derivative of Y with respect to X is equal to 10x multiplied by E raised to the quantity of 5 x squad. Now we're asked to identify the inner and outer functions, or our original function Y is equal to E raised to the quantity of 5 X2 + 21. So, for our inner function U, it needs to be a function of X. So we're actually going to choose the exponent of our exponential as our inner function. So that means that you are going to be equal to 5 X squad. Plus 21. And so that is what we're going to identify as our inner function. So now that we have our inner function identified, we can rewrite our function Y, which will now be our outer function, since Y is going to be a function of you. And this is going to be equal to E raised to the U. And so that is going to be our outer function. So now that we have the inner and outer function identified, we can look at the derivative of Y with respect to X, and to calculate this derivative, we're going to use 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 will use our inner and outer functions to calculate these derivatives. So, the first derivative that we'll calculate is the derivative of U with respect to X. This is going to be equal to the derivative with respect to X. Of the quantity of 5 X2 + 21. Now, in order to take this derivative, we need to recall a couple of rules. The first one is gonna be our power rule that says that the derivative with respect to X of the quantity of X raised to the N power is going to be equal to N multiplied by x raised to the N minus 1. We will also need to take the derivative of a constant, and so we recall that for a derivative of a constant, we'll have the derivative with respect to X of a constant C, and that's going to be equal to 0. So applying these rules to our derivative of U with respect to X, this is going to be equal to. 5 multiplied by 2, which is going to be 10, multiplied by X, and that's gonna be our derivative. Since when we take the derivative of 21, that's a derivative of constant, so it's derivative will be 0. We now need to also take a 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 E rays to the U. So, recall your rules for taking a derivative of an exponential, that is the derivative with respect to X of the quantity of E to the X is equal to E X. So we get back just the same function. That means that our derivative of Y with respect to U is just going to be equal to E rates to the U. So now that we have both of our derivatives calculated, we can substitute the results back into our chain rule. So we'll have the derivative of Y with respect to X is equal to the derivative of Y with respect to U, which we just calculated to be E to the u. However, we want our final answer to be a function of X. So for you, we're going to replace that with the 5 X2 + 21. So this means I'm going to have E raised to the quantity of 5 X2 + 21. And then that gets multiplied by Our derivative of you with respect to X, which we calculated earlier to be 10 X. So we'll multiply this quantity by 10 X. So just rewriting this. In the normal fashion, we will have 10 X, multiplied the uh E raised to the quantity of 5 X2 + 21. And so that is the answer for a derivative that we were looking for. And now, if I take the inner and out functions that I've identified, as well as the result of our derivative of Y with respect to X, and look at our answer choices, the correct answer choice corresponds to answer choice C. 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 = e^4x²+1

Key Concepts

Composite Functions

Chain Rule

Exponential Functions

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