Derivatives Find and simplify the derivative of the following ...

Question

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

Accepted Answer

Hello. In this video, we are going to be calculating the derivative of the given function. We are given a function F of X. Which is defined as 2 X divided by the quantity 3X minus 4. So, in order to take the derivative of this function, we will need to use the quotient rule for derivatives. The quotient rule states that if we try to take the derivative of a rational function such as F of X divided by G of X, then the derivative will be defined as the denominator multiplied by the derivative of the numerator minus the numerator. Multiplied by the derivative of the denominator, all divided by the denominator squared. So, we will be treating the numerator as a function F of X and the denominator as the function G of X. And as long as we have those defined functions and their derivatives, we can plug them in to the derivative of the quotient role. So, let's go ahead and start by defining our F of X and G of X for the quotient rule. F of X will be the function located in the numerator of our given function. That is going to be 2 X. So in our case, F of X will equal to 2 X. Next, G of X will be the denominator, and that is going to be 3X minus 4. Now we need to define their derivatives. By using the power rule, the derivative of the numerator F of X will equal to 2. By using the power rule and constant rule, the derivative of the denominator G of X. Will equal to 3. So now that we have the 4 pieces of information we need, we can go ahead and plug it into the definition of the quotient rule to find the derivative of the original FFX function. So again, by using the quotient rule, the derivative will be defined as the denominator. 3 X minus 4 multiplied by the derivative of the numerator, which is 2, minus the original numerator, which is 2 X, multiplied by the derivative of the denominator, which is 3, all divided by the original denominator squared, 3 X minus 4 quantity squared. Now that we have plugged in all the information we need, all we need to do is algebraically simplify. We can go ahead and start by distributing positive 2. Into the quantity 3 X minus 4. And 2 X multiplied by 3 will leave us with 6 X. So, what we have is 6 X minus 8. Minus 6 X, all divided by 3 X minus 4 quantity squared. In the numerator, we can combine the six X terms together to cancel them out. That will leave us with negative 8 divided by 3 X minus 4 quantity squared. This is the simplest form of our derivative, so F of X will equal to -8 divided by 3X minus 4 squad. And that is going to be the derivative of the original of the original 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 on how to use the quotient rule, and I'll go ahead and see you all in the next video.

Derivatives Find and simplify the derivative of the following functions.
f(x) = x /x+1

Key Concepts

Derivatives

Quotient Rule

Simplification of Derivatives

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