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.

f. (x¹¹ + f(x))⁻², x = 1

Accepted Answer

Welcome back, everyone. In this problem, given that F of -1 equals -1 and F of -1 equals -2, find the derivative of X 9 plus 2 FX raise to the power of -3 at X equals -1. A says it's -5/27, B 5 divided by 81, C-59, and D 5/3. Now, in this problem we want to find the derivative of this expression and then evaluate it at this point at X equals -1. How can we do that? Well, notice here that our expression is raised to the power of -3. It's a composite function and to differentiate composite functions we can use the chain rule. Recall that by the chain rule of differentiation it says the derivative of a composite function f of G of X. Is going to be equal to the derivative of the outer function F of G of X. Multiplied by the derivative of the inner function G of X. So if we can apply the chain rule, we can find the derivative of our expression and then evaluate it at X equals -1. By applying that rule, then that means our derivative. OK. Is gonna be equal to. First we bring down the po, so we're differentiating the alter function. So that's -3 multiplied by X to the 9th plus 2 F of X, OK, raised to the power of -3 minus 1, multiplied by the derivative of the inner function. So that's the derivative of X 9 plus 2 F of X. Now here for our auto function this is gonna be -3 multiplied by X 9 plus 2 F of X raise to the power of -4, and the derivative of our inner function is going to be 9 x 8. OK, that's the derivative of X 9th by the power rule plus 2F X. That would be the derivative of 2F X. And now when we simplify here, then we get this to be negative 3 multiplied by 9 X to the 8 plus 2 F of X. All dividing, OK. All divided by X to the 9th plus 2 F of X raised to the 4th, OK? We applied what we knew about powers. Knowing that we have our derivative, we can evaluate it at X equals -1. But remember earlier we were told that F of -1 equals -1 and F-1 equals -2. And this is going to come in handy because at X equals -1, OK, then. Or or or function is gonna be -3 multiplied by 9 X. To the 8 or 9 multiplied by -1 to the 8th, OK, plus 2 F of X, so 2 F of -1. All divided by -1 to the 9th, OK, plus 2 multiplied by F of -1. Sorry, yeah, yeah, 2 multiplied by F-1 to the 4th. Now, when we go ahead and simplify that, that's gonna be -3, OK, multiplied by 9, where to the point of -1 to the 8th, which is going to be 1. Plus 2 multiplied by F-1, so that's 2 multiplied by -2 all divided by -1 to the 9th, that's -1 plus 2 multiplied by F-1 that which would be -1, OK, always to the 4th. Now when we start to simplify here, so that's gonna be -3, multiplied by 9 minus 4, it's multiplied by 5, divided by -1 minus 2 as -3 to the 4th. Now, we can simplify here we can cancel out one of the negative 3s in our denominator, leave it as 5 divided by -3 cubed. 3 cubed is -27 and 5 divided by -27 is -527. Thus, the value of our derivative at X equals -1 is -527. If we look back at our answer choices, that means A 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.

f. (x¹¹ + f(x))⁻², x = 1

Key Concepts

Chain Rule

Power Rule

Derivative of Inverse Functions

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