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(s) = √s/4

Accepted Answer

Hello. In this video, we are going to be determining the derivative of the function H of X. Now, the problem tells us that the function H of X is equal to 5 multiplied by the square root of X divided by 6. Before we start taking the derivative, let's go ahead and rewrite H of X in an exponential form. So the first thing we're gonna do is we're going to rewrite. Are given function as a product of a constant and a variable. We can rewrite 5 multiplied by the square root of X divided by 6, as 5/6 multiplied by the square root of X. Next, we're going to rewrite the variable X in an exponential form. The square root of X can be rewritten as X to the power of 1/2. So we can further rewrite H of X as 5/6 multiplied by X to the power of 1/2. Now, how can we take the derivative of this exponential function? While in order to take the derivative, we will be using the exponential rule of the derivative. The rule states that if we have some function in exponential form, let's say F of X equal to x the power of n, the derivative will equal to the exponential value n moved to the front of the variable. So, we have N multiplied by the variable X, then we will decrease the exponent N by 1. So the derivative of an exponential function is of the form and multiplied by X the power of N minus 1. We will use this to take the derivative of H of X. Now, our exponential function in this case is going to be X to the power of 1/2 of 1/2. In this case, our N is going to be 1/2. Now The derivative H of X is going to equal to 56 multiplied by the derivative of X to the power of a half. That will be 1/2 multiplied by X raised to the power of 1/2 minus 1. That is the derivative of the exponential value x to the power of the half. Now what we need to do is we need to simplify. Let's start by simplifying the coefficients. 5 divided by 6 multiplied by 1/2 will simplify to give us 5 divided by 12. And the exponent 1/2 minus 1 will simplify to be negative 1/2. So we have X raise the power of -12. Now, we don't want our solution to have a negative exponent, but what we can do is we can rewrite or move the variable X, the denominator, in order to make it a positive exponent. So this will be equivalent to 5 divided by 12 multiplied by X to the power of positive half. Finally, we will go ahead and rewrite X to the power of 1/2 back in a square root form. So the final form of H X is going to be 5 divided by 12 multiplied by the square root of X, and this is going to be the derivative of our HX function. And with that being said, the solution to this problem is going to be C. So, I hope this video helps you in understanding how to use the power rule for the derivatives, and I'll go ahead and see you all 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(s) = √s/4

Key Concepts

Derivatives

Power Rule

Chain Rule

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