9–61. Evaluate and simplify y'.

y = (2x−...

Question

9–61. Evaluate and simplify y'.

y = (2x−3)x^3/2

Accepted Answer

Hello. In this video, we are going to be calculating the derivative of the following function. We are given the function Y is equal to the quantity 4 X minus 5 multiplied by X rates to the power of 1/2. So, notice that the function Y is given to us as a product of two functions of X. In order to calculate the derivative, we can go ahead and use the product rule for the derivative. The product rule states the following. If we want to take the derivative of the product of two functions, F of X and G of X, then the derivative is defined as F of X, multiplied by the derivative of G of X plus G of X multiplied by the derivative of F of X. So, what we need to do is we need to define our F of X and G of X function in order to use the product rule. We will define the quantity 4 X minus 5 as our F of X function. And we will define X rates the power of 1/2 as our G of X function. Now what we need to do is we need to take the derivatives of F of X and G of X in order to use the product rule. By using the power rule, the derivative F of X will equal to 4. And by using the power rule on G of X, the derivative G of X will equal to 1/2 multiplied by X raise to the power of -12. Now that we have the 4 pieces of information we need, we will go ahead and plug these values into the derivative of the product rule. So The product rule or the derivative will be defined as the following. It will be F of X, which is 4 X minus 5, multiplied by the derivative of G of X, which will be 1/2 multiplied by X rates the power of negative 1/2, plus G of X, which is X rates the power of 1/2 multiplied by the derivative of F of X, which will be 4. Now what we need to do is we need to go ahead and simplify. What we will do is we will start by distributing 1/2 X raises the power of negative 12 to 4 X minus 5. And X rates the power 1/2 multiplied by 4, will leave us with 4 X rates the power of a half. So our derivative will simplify to be 2 multiplied by X rates the positive half power, minus 5 divided by 2 multiplied by X rates the negative half power, plus 4 multiplied by x raised to the half power. Next, what we will do is we will go ahead and combine light turns. 2 X rates to the half power, and 4 X rays to the half power will combine to be 6 X rays to the half power. So our derivative will simplify to be 6 X rays to the power minus 5 divided by 2 multiplied by X-rays to the negative half power. Finally, we don't want to leave any negative exponents in our solution. So we will move X rates to the negative half power to the denominator of 5 divided by 2 in order to make it positive. This will allow us to rewrite our derivative as 6 X rates to the half power minus 5 divided by 2 multiplied by X-rays to the power. This is going to be the simplest form of the of the product rule, and this is going to be our derivative. Furthermore, the solution to this problem is going to be a. So, I hope this video helps you in understanding on how to use the product rule, and I will go ahead and see you all in the next video.

9–61. Evaluate and simplify y'.

y = (2x−3)x^3/2

Key Concepts

Differentiation

Product Rule

Simplification

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