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 = (3x+7)¹⁰

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. So 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, then find the derivative of Y with respect to X, and we're given the function of Y is equal to the quantity of 4 x minus 1 in quantity rates to 6. We have 4 possible choices as our answers. For choice A, we have U is equal to 4 x minus 1. F of U is equal to U 6, and our derivative of Y with respect to X is equal to 24, multiplied by the quantity of 4 X minus 1 in quantity rates to the 4th. For choice B, we have U is equal to 4 X minus 1, F of U is equal to U 5th, and the derivative of Y with respect to X is equal to 20, multiplying the quantity of 4 X minus 1 in quantity rates to the 4th. For choice C we have U is equal to 4 x minus 1, F of U is equal to u raised to 6, and our derivative of Y with respect to X is equal to 6, multiplying the quantity of 4 X minus 1 in quantity raised to the 5th. And finally for choice D, we have U is equal to 4 x minus 1, F of U is equal to u raised 6th, and our derivative of Y with respect to X is equal to 24, multiplying the quantity of 4 x minus 1 in quantity rates to the 5th. So this question asks us to identify the inner and our function for composition function Y, as well as take the derivative of Y with respect to X. Let's begin by identifying our inner and outer functions. Now for in function, we need U, which is a function, which is equal to G of X. So we need you to be a function of X. And so, if you look at our function Y, we can choose you to be the quantity inside of the parentheses. That means we're gonna have you. Equal to 4 X minus 1. And so that's gonna be our inner function. And so if we said U equal to 4 X minus 1, that means that Y, which is equal to F of you, which is going to be our outer function, is going to be equal to You race to the 6th. So, those are the two identifications that we needed to make. So now we need to use this information to calculate the derivative of Y with respect to X. So, to do this, we need to recall our 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. So we're gonna need to take a derivative of you with respect to X and a derivative of Y with respect to you. And so we can use, we can use our inner and outer functions to take those derivatives. So taking a derivative of you. With respect to X, That's gonna be equal to the derivative with respect to X of the quantity of 4 X minus 1. And when we take this derivative, the derivative of 4 X is just 4, and the derivative with respect to X of minus 1 is going to be 0. So this evaluates to just 4. We also need to take the derivative of Y with respect to you. So here we will be using our outer function. So this is going to be equal to the derivative with respect to you, of the quantity of you raised to the 6th. And when I take this derivative, it's going to be equal to 6 multiplied by U raised to the 5th. So recall when you take a derivative of a um polynomial that we're going to multiply our U by the exponent, and then subtract one from the exponent. So we get um the derivative of U rated to the 6 is equal to 6 multiplied by U rate to the 5th. So now we can take these two quantities and substitute them back into our formula for the chain rule. So we'll have the derivative of Y with respect to X. It's going to be equal to the derivative of Y with respect to U, and that we calculated is 6 multiplied by U raised to the fifth. We'll have 6 multiplied by, and here we're going to go ahead and substitute back in for you for X minus 1, since we want our final answer to be a function of X. So we'll have 6 multiplying the quantity of 4 X minus 1 in quantity raised to the fifth. Then this gets multiplied by our derivative of U with respect to X, which we calculated to be 4. So this quantity gets multiplied by 4. And so we can just simplify this expression by multiplying the 4 by 6. And this will give us 24, multiplying the quantity of 4 X minus 1 in quantity raised to the fifth. And so that is the answer for the derivatives that we are asked to calculate. And so now that we have our two functions identified, as well as the derivative calculated, 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 = (3x+7)¹⁰

Key Concepts

Composite Functions

Chain Rule

Differentiation

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