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.

b. f(x)g³(x), x = 0

Accepted Answer

Welcome back, everyone. Given that F of -1 equals -1, F-1 equals -2, G of -1 equals -1, and G of -1 equals 5, find the derivative of F of X to the 4th multiplied by G of X at X equals -1. A says it's -8, B5, C-3, and D1. Now for this problem we'll need to first find the derivative of this expression and then evaluate it at this value of X X equals -1. Now if you take a good look at the expression here, we're multiplying one term by the other, and one of these terms is a function that's been raised to a power. So to differentiate it we'll need the chain and the product rules of differentiation. What do they say? Well, recall that the chain rule of differentiation says that if we're finding the derivative of a composite function F of G of X, OK, then it's going to be equal to the derivative of the other function F of G of X, OK, multiplied by the derivative of the inner function G of X. Next, we also know that the product rule of differentiation says that if we're differentiating two terms or if we're differentiating our product, OK, FFX multiplied by GFX. Then its derivative is going to be G of X multiplied by the derivative of F of X. Plus affix multiplied by the derivative of geofi. In other words, we keep one term and differentiate the other. So now that we have both of these rules, let's apply them to find the derivative of our expression. Now if we take it here, that means the derivative of FFX to the 4th multiplied by GFX. Applying the chain rule is gonna be equal to geoX multiplied by the derivative of FFX. So that's the derivative, sorry, the derivative of our first part of our term. So that's the derivative of FFX to the 4th because that's what our term is, OK. Plus You know, we keep in this case, we keep the 1st known and differentiate the second, so FFX to the 4th multiplied by the derivative. Of GFX. Now what are those derivatives going to be? Well, here. To differentiate F of X, and let me, let me make some space down here, to differentiate F of X to the 4th, we can apply our chain rule. So now this is gonna be G of X multiplied by 4 F rates to the 4 minus 1 of X, OK. That's the derivative of the auto function multiplied by the derivative of our inner function. That would be the derivative of F of X or F of X, OK? Plus F of X to the 4th multiplied by the derivative of G of X, which is, well, here, which is gonna be G of X, OK? And now if we simplify this, OK, then we're gonna get 4 FFX cubed multiplied by G of X, multiplied by the derivative of F of X, OK. Plus F of X to the 4th multiplied by the derivative of G of X, OK? So this is gonna be the derivative of our expression. Now we need to evaluate it at X equals -1. Now recall earlier we said that F of -1 equals -1, F-1 equals -2, G of -1 equals -1, and G -1 equals 5. And this is gonna be helpful because at X equals -1, OK, then. Our derivative is gonna be 4 F of -1 cubed, OK. Multiplied by G of -1 multiplied by F-1 plus F of -1 to the 4th multiplied by G of -1 and now we can substitute our known values. So this is going to be 4 multiplied by -1Q because f of -1 is -1. Multiplied by G of -1, which is -1 multiplied by F-1 2 plus F of -1 to the 4th, so that's -1 to the 4th multiplied by G-1, which is 5. Now when we go ahead and simplify here, -1 cubed is -1 and -1 multiplied by -1 multiplied by -2 will be -2.1 multiplied by -2 is -8. On the other hand, -1 to the 4th is 1 and 1 multiplied by 5 is 5. So now we have negative 8 + 5, which equals -3. Thus, at X equals -1, the derivative of our function is -3. If we look back at our answer choices, that means that 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.

b. f(x)g³(x), x = 0

Key Concepts

Product Rule

Chain Rule

Substitution of Values

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