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.

h(t) = t²/2 + 1

Accepted Answer

Hi, everyone, let's take a look at this practice problem dealing with derivatives. This problem says to calculate the derivative of the function G of X, which is equal to X cube divided by 3 + 5. We're given 4 possible choices as our answers. For choice A, we have X cubed. For choice B, we have X2d. For choice C we have 3 X2, and for choice D, we have 2 X cubed. Now we're asked to calculate the derivative of our function G of X, so we're going to be looking at our function G of X. So here I'm gonna be taking a derivative of G with respect to X. This is going to be equal to the derivative with respect to X of the quantity of X cubed divided by 3 + 5. Now, to take a derivative of this function, we're going to apply several of our derivative rules. So, the first one is going to be our constant multiple and power of rule. And so recall for that, we have the derivative with respect to x of the quantity of C multiplied by x raised to the power n, and this is going to be equal to C multiplied by N, multiplied by x raised to the quantity of n minus 1, where C here is a constant. The other rule that we're gonna need to use is our constant rule, and that says that the derivative with respect to X. Of a constant D. It's going to be equal to 0. So we're gonna have to apply both of these rules to our derivative. So coming back to our derivative, the first term, when I take a derivative of X cube divided by 3, here I'm going to need to use our constant multiple and power rule. So here in this case, C is going to be equal to 1 divided by 3, and N is going to be equal to 3. So I want to take the derivative of our first term, we're going to have 1 divided by 3, multiplied by 3, multiplied by X, raised to the quantity of 3 minus 1. Then we'll have plus for our second term, I'm taking a derivative of a constant, which is 5. And when I take a derivative of constant due to our constant rule, it is going to be 0, so we'll have plus 0. So now we just need to simplify this expression and do it so this is going to be equal to x2d. Since 3 multiplied by 1 divided by 3 gives me a coefficient of 1, and then in the exponent I have 3 minus 1, which is 2. So this is our uh derivative that we were looking for. And if we compare our final answer here to our answer choices, the correct answer choice corresponds to answer choice B. 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.
h(t) = t²/2 + 1

Key Concepts

Derivatives

Power Rule

Function Notation

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