Suppose that functions ƒ(x) and g(x) and their first ...

Question

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

g. ƒ(x + g(x)), x = 0

Accepted Answer

Hello. In this video, we are going to be calculating derivative of the given function by using the table. We are asked to find the derivative of G X plus F of X, when X is equal to -1, and we want to find the value of the derivative by using the given table. First, let's go ahead and find the derivative of the given function. We are given the function G of X plus F of X. This is in the form of a composite function, so if we want to take the derivative of a composite function, we will need to use the chain rule. To start, we will take the derivative of the outermost function, and the outermost function is going to be the function G. The derivative of G will be G, and by the chain rule, we will keep the innermost function the same. So we have X plus F of X inside the function G. Then by the chain rule, we will multiply the outermost derivative by the derivative of the innermost function. The innermost function is X + F of X. So we will go ahead and multiply G by the derivative of X plus F of X. The derivative of X with the power rule is going to be 1, and the derivative of F of X will be F of X. So the derivative of the given function will be G X plus F of X multiplied by 1 + F of X. Now we want to evaluate this derivative when X is equal to -1. We will start by plugging in -1 into the composite function or into the derivative. So, that will give us G of -1 plus F of -1. Multiplied by 1 plus F of -1. Now what we want to do is you want to find the values of the independent functions. Let's go ahead and use the table to find the value of F of -1. We will take a look at the row that has the values of the function F of X. When X is equal to -1, this function has the value of 3. So we will replace F of -1 with 3 by using the table. Then we will find the value of of -1. We will take a look at the row with the values of the derivative of F of X, and when X is equal to -1. F of X has the value of 2. So we will replace F of -1 with 2 by using the table. That will allow us to rewrite the function to be G of -1 + 3 multiplied by 1 + 2. And this will further simplify to be G 2 multiplied by 3. Now, what we need to do is we need to use the table again to find the value of G 2. So we will take a look at the row with G. When X is equal to 2, G has a value of 6. So we will replace G of 2 with 6 by using the table, and that will allow us to simplify the function to be 6 multiplied by 3, which is 18, and this is going to be the solution to the given equation. And the overall solution is going to be B. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

g. ƒ(x + g(x)), x = 0

Key Concepts

Chain Rule

Product Rule

Evaluating Derivatives at Specific Points

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