Derivatives of products and quotients Find the derivative of t...

Question

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.

h(x) = √x (√x-x³/²)

Accepted Answer

Hello. In this video, we are going to be computing the derivative of the given function after simplifying the function, and we want to provide the simplified answer. The function given to us is F of Z equal to Z raised to the 1/5th multiplied by the quantity, Z 1/5 minus Z 9/5th power. So, before we take the derivative, let's go ahead and simplify the function. In order to simplify the function, the first thing we can do is distribute Z 1/5 power into the multiplied quantity. That will allow us to rewrite the function as F of Z equal to Z. Z 2/5 power. Minus 2 to the 105th power. In the last term, 10 divided by 5 will simplify to be 2, so we can rewrite the last term as Z squared. This is going to be the simplest form of the given function, so let's go ahead and take the derivative. In order to take the derivative, we are going to have to use the power rule of the derivatives, since each of the terms of Z are in an exponential form. So, the power rule states the following. If we have a function in the exponential form, such as X raises the power of N, the derivative rule states that we will take the value of N and move it to the front of the X variable. Then we will decrease the value of the exponent by 1. So, let's go ahead and apply the power rule to each individual term of F of Z. So we will call the derivative F of Z. Now, for the first term, we have an exponential value of 2/50. So what this means is that our N value, by the power rule is going to be 2/5 for the first term. By the power rule this derivative will be 2/5 multiplied by Z raised to the power of 25 minus 1. Then we will repeat this process for the last term. Now, we have Z2d and in this term, our N value for the power rule is going to be 2. So the derivative of the last term will be 2 multiplied by Z, raised to the power of 2 minus 1. Now what we want to do is you want to go ahead and simplify the derivative. In the first exponent, 2/5 minus 1 will give us the value of -35. So we are left with 2/5 multiplied by Z raise the power of -35. And in the final term, the exponent 2 minus 1 will simplify to be 1, and that will leave us with 2 multiplied by Z. Now, what we have here is a negative exponent in the first term. Since we want to give the problem the simplified answer, we want to go ahead and Return or change this negative expo into a positive one. In order to do this, we can take the term Z raise the power of -35s and move it to the denominator of the coefficient. That will allow us to rewrite the derivative as F Z equal to 2 divided by 5 multiplied by Z raised to the positive 3/5th power. Minus 2 Z. and this is going to be the simplified form of the derivative of the given function. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding on how to use the power rule, and I will go ahead and see you all in the next video.

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.
h(x) = √x (√x-x³/²)

Key Concepts

Product Rule

Quotient Rule

Simplification of Expressions

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