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.

d. ƒ(g(x)), x = 0

Accepted Answer

Hello. In this video, we are going to be taking the derivative of the given function. So we're asked to find the derivative of G of F of X, when X is equal to -2 by using the table below. The table gives us different values for the functions F of X and G of X, and their derivatives at various values of X, going from -1 to 2. So, in order to start this problem, let's just first write down the given function. We are given G of F of X, and what we want to do is we want to take the derivative of this function with respect to X. In order to take the derivative of a composite function, we will need to use the chain rule. We will start by taking the derivative of the outermost function, which is going to be G. The derivative of the function G will be G. Then, by the chain rule, we will go ahead and keep the innermost function the same, so we have G of F of X. Then we need to multiply this function by the derivative of the innermost function. The innermost function is F of X and F of X's derivative will be F of X. So this is going to be the derivative of G of F of X, and now we want to evaluate this derivative when X is equal to -2. In order to do this, we will plug in the value of -2 into the derivative. That will give us G of F of -2. Multiplied by F of -2. In order to find the values of this function, we are going to use the table above. Let's go ahead and start by finding the value of F of -2. In order to find the value of F of -2, we will look at the row of the values of F of X. Then we want to find the value of F of X at -2. And this value is going to be positive too. So we know that the value of F of -2, based off the table, will be 2. Now we need to find the value of F of -2. By using the table, we will take a look at the row of values for the derivative F, and we will look at the value when X is equal to -2. That value is going to be 5. So F of -2 will be 5. So replacing these values into the function will leave us with G of 2 multiplied by 5. 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 values, so derivative G, and when X is equal to 2, the value of this function is going to be 6. So, we will replace G with 6, leaving us with 6 multiplied by 5, and this product will be 30. So this is going to be the value of the derivative G of F of X, and the solution to this problem is going to be A. So I hope this video helps you in understanding on how to approach this problem, and I'll 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.

d. ƒ(g(x)), x = 0

Key Concepts

Chain Rule

Function Values and Derivatives

Evaluating Derivatives at Specific Points

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