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 = √7x-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 for the composition of Y is equal to F of G of X. Then define the derivative of Y with respect to X. We're given the function. Y is equal to the square root of the quantity of 8 X minus 7. We have 4 possible choices as our answers. For choice A, we have U is equal to 8 X minus 7. F of U is equal to u raised to the 1/3, and the derivative of Y with respect to X is equal to 8 divided by the square root of the quantity of 8X minus 7. For choice B, we have U is equal to 8 x minus 7. F of U is equal to u 13, and the derivative of Y with respect to X is equal to 4 divided by the square root of the quantity of 8 X minus 7. For choice C, we have U is equal to 8 x 7. We have F of U is equal to 12, and the derivative of Y with respect to X is equal to 8 divided by the square root of the quantity of 8 XY 7. And for choice D, we have U is equal to 8 x minus 7, F of U is equal to u 12, and the derivative of Y with respect to X is equal to 4 divided by the square root of the quantity of 8 X minus 7. Now, the first thing we need to do in this problem is to identify our inner and outer functions. We're told that our inner function U has to be a function of X. So for our inner function, we're going to choose the quantity that is underneath the radical in our equation. So that means that U is going to be equal to 8 X minus 7. So that is our inner function identified. And now that we've identified our inner function, we can rewrite our function Y. So, Y is now going to be a function of you. And when I replace 8 X minus 7 with U and our equation, this is going to be equal to U raised to the 12 power, since the square root of a quantity is the same as raising that quantity to the 12 power. So this is going to be our outer function. So now that we have our inner and outer functions identified, we can now calculate the derivative of Y with respect to X. And for this, we're going to need to use our chain rule. So recall for the chain rule that the derivative of Y with respect to X is going to be equal to the derivative of Y with respect to U. Multiplied by the derivative of u with respect to X. And so what we're going to do is use our inner and outer functions to calculate these derivatives. So, the first derivative that I'm going to 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 8 X minus 7. Now, to take this derivative, we need to remember two rules for our derivatives, and that's gonna be our power rule, which is the derivative with respect to X of the quantity of X to N, and that is equal to N multiplied by x raised to the N minus 1. We're also gonna be taking a derivative of our constant, so we need to remember our rules for derivatives with constants. That's gonna be the derivative with respect to X of a constant which we'll label as C is going to be equal to 0. So we can use these two rules to calculate our derivative of U with respect to X, and in doing so, this derivative is just going to be equal to 8. Now we need to also calculate the 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 raised to the 12 power. So, here again, we will apply our power rule for derivatives, and this is going to be equal to 1/2 multiplied by U raised to the minus 12 power. So now I have both of these derivatives calculated, I can substitute their results back into our formula for the chain rule. So that means that the derivative of Y with respect to X is going to be equal to the derivative of Y with respect tou, which is 1/2 multiplied by you raised to the minus 12. However, we want our final answer as a function of X. So for you, we're going to substitute in 8 x minus 7. So this quantity is going to be equal to 12, multiplying the quantity of 8 X minus 7 in quantity raised to the minus 12 power. We then need to multiply this by the derivative of U with with respect to X, which we calculated to be 8. So we'll multiply this quantity by 8. So now all we need to do is simplify our expression. So we will have the derivative of Y with respect to X is equal to, and here we'll have 8 divided by 2, which will give me 4. And that's gonna be divided by the square root of the quantity of 8X minus 7. So, here we've written the quantity of 8X minus 7, raised to the in quantity, raised to the minus 1 half hour. As one divided by the square root of the quantity of 8 X minus 7. So this is the answer that I was looking for for my um value for my derivative. And so now that I have identified both the inner and outer function, as well as calculate the derivative, 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 = √7x-1

Key Concepts

Composite Functions

Chain Rule

Inner and Outer Functions

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