Suppose that the functions f and g and their derivatives with ...

Question

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.

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Find the derivatives with respect to x of the following combinations at the given value of x.

g. f(x + g(x)), x = 0

Accepted Answer

Welcome back, everyone. In this problem, given that F of 1 E equals -1, F1 equals 3, G 1 E equals 0, and G 1 equals 4, find the derivative of F of X plus 3XG X sorry, at X equals 1. A says it's negative 13, B 13, C 39, and D 39. Now how are we gonna find the derivative of this function? We'll notice that it's a composite function so we can apply the chain rule. Recall that by the chain rule, it says that the derivative of a composite function f of GFX. Is going to be equal to the derivative of the auto function F of G of X multiplied by the derivative of the inner function G of X. So if we can apply that idea here, we can find the derivative of our expression and then evaluate it at X equals 1. Because no, that means that the derivative. Of F of X + 3 G of X. OK. It is going to be equal to the derivative of the other function, so that's F of X + 3G of X, OK. Multiplied by the derivative of our inner function, which is X + 3 G of X. I know when we differentiate the inner function, OK, let me just write back our first derivative. It's gonna be equal to, OK, uh, 1 derivative of X is 1 plus 3G X, which would be the derivative of 3G X. Since we have the derivative, where do we go from here? Well, no, remember that we were told. Earlier in our problem that F of 1 equals -1, F1. Equals 3 G of 1 equals 0. And G of 1 equals 4. And that's going to be, that's going to come in handy because at X equals 1, then our derivative is going to be F. Of 1 + 3G of 1, OK, multiplied by 1 + 3G of 1, OK? I know, if we, if we start to simplify here, that means we're gonna have F of 1 + 3 multiplied by 0 because G of 1 is 0, multiplied by 1 + 3 multiplied by G of 1, which is 4, OK? Now in this case We're gonna have F of 1 + 3 multiplied by 0, that's gonna be 1, OK, multiplied by 1 + 12. Now if we think about it here, like we said, the derivative of F1 or F 1 is 3, so this is gonna be 3 multiplied by 13, which is gonna equal 39. Thus, the derivative of our expression at X equals 1 is 39. If we look back at our answer choices, that means C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.

Find the derivatives with respect to x of the following combinations at the given value of x.

g. f(x + g(x)), x = 0

Key Concepts

Chain Rule

Function Composition

Derivative Evaluation

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