Derivatives Find the derivative of the following functions. Se...

Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

f(v) = v¹⁰⁰+e^v+10

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with derivatives. This problem says to determine the derivative of the function PFT, which is equal to 4 multiplied by T rated to the 8th, plus 3 multiplied by E raised to the T minus 5. We have 4 possible choices as our answers. For choice A, we have 32 multiplied by T raised to the 7th, plus 3 multiplied by E raised to the T minus 5. For choice B, we have 32 multiplied by T raised to the 7th, plus 3. For choice C, we have 4 multiplied by T raised to the 8th, plus 3 multiplied by E raised to the T, and for choice D, we have 32 multiplied by T raised to the 7th, plus 3 multiplied by E raised to the T. So we're asked to take a derivative of our function of PFT. So that means we're gonna be looking for P of T. And so that is going to be a derivative with respect to T of our function of PFT. So we'll take a derivative with respect to T of the quantity of 4 multiplied by T rated to the 8th, plus 3 multiplied by E raised to the T minus 5. So, in order to take the derivative of this expression, we're gonna need to recall several of our derivative rules. So the first rule that we're gonna need to use in the first term is going to be our power rule. So recall for the power rule that we're gonna have the derivative with respect to T. Of the quantity oft raised to the end. And this is going to be equal to N, multiplied by T raised to the N minus 1. Now, for the second term, we're gonna need to use our um derivative rule for exponents to recall for that rule that we have the derivative with respect to T of the quantity of E raised to the T, and that is going to be equal to E raised to the T. Now, for the last term, we're going to need to use our uh constant derivative rule. And so, that is going to be the derivative with respect to T of a constant C is going to be equal to 0. And finally, we're going to need to use our constant multiple rule for the first two terms. As you recall for that rule that we have the derivative with respect to T of the quantity of C multiplied by F of T. And this is going to be equal to C multiplied by the derivative with respect to T. Of F of T. So now we're going to apply all these rules to our function. And so, coming back, this is going to be equal to. When we take the derivative with respect to T of 4, multiplied by T rates to the 8th, by the constant multiple rule, the 4 will move through the derivative, then I'll just have the derivative with respect to T of T rates to the 8th, and there we can use our power rule, where N is equal to 8. So, for the first term, we're gonna have 4, multiplied by the exponent, which is 8. Multiplied by T, raised to the quantity of 8 minus 1. Then we'll have plus for the 2nd term, we're again going to use the constant multiple rule. So we'll have the 3 move through the derivative and then we'll just have the derivative of E raised to the T. And so we take the derivative of E raise to the T, with respect to T, we get back E raises to the T. So for a second term, we'll have plus 3 multiplied by E raised to the T. And then finally, our last term, we're taking a derivative of a constant, which is -5. And so applying our constant derivative rule, that it's going to be 0, so we'll have plus 0. So now we just need to simplify this expression. So this is going to be equal to, we'll have 4 multiplied by 8, which is 32. That's gonna be multiplied by T raised to the 7th, so it's 8 minus 1 is 7. Then we'll just have plus 3 multiplied by E raised to the T. And so, this is the answer that we are looking for. And if we compare our answer to our answer choices, the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(v) = v¹⁰⁰+e^v+10

Key Concepts

Derivatives

Power Rule

Exponential Functions

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