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

Question

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.

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

f. √f(x), x = 2

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given that F of -1 is equal to 3 and F-1 is equal to -6, find the derivative of 3 multiplied by the square root of F of X at X equals -1. So it appears for this particular problem, we're ultimately trying to figure out the derivative of 3 multiplied by the square root of F of X at X equals -1, provided the information that is given to us by the prom itself. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is negative square root of 3, B is positive square root of 3. C is 3 multiplied by the square root of 3, and finally D is -3 multiplied by the square root of 3. Awesome. So first off, let us recall that D by DX of the square root of X is going to be equal to 1 divided by 2 multiplied by the square root of X. Let us also recall and use the chain rule which states that D by DX of F of G of X. is equal to F of G of X multiplied by G of X. So let us apply the chain rule to help us find the derivative of 3 multiplied by the square root of FFX. So when we do that, we will get that D by DX of 3 multiplied by the square root of FFX. It's going to be equal to 3 multiplied by 1 divided by 2 multiplied by the square root of F of X. Multiplied by F of X, which will be equal to, so it will be all equal to 3 multiplied by F of X divided by 2 multiplied by the square root of F of X. Fantastic. So now we need to evaluate the derivative at X equals -1, and we also need to use the given values of F. Of -1 is equal to 3 and F of -1 is equal to -6. So we can go ahead and write that 3 multiplied by F of X, divided by 2 multiplied by the square root of F of X is going to be equal to. And don't forget that we are evaluating this at X equals -1, so this will be equal to. 3 multiplied by F-1 divided by 2 multiplied by the square root of F of -1, which will be plug in or known variables will be equal to 3 multiplied by -6, divided by 2 multiplied by the square root of 3. And when we multiply this or when we perform this we perform this calculation in our calculators, so when we perform this valuation in our when we plug this into our calculator, so if we plug in 3 multiplied by -6 divided by 2 multiplied by the square rooted 3, and we write it in its most simple form, we will find that it's equal to -3 multiplied by the square root of 3, and that will be our final answer. Hooray, we did it, and this corresponds to multiple choice answer letter D, which once again is -3 multiplied by the square root of 3. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.

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

f. √f(x), x = 2

Key Concepts

Derivative

Chain Rule

Square Root Function

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