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

Question

Derivatives Find and simplify the derivative of the following functions.

f(x) = 2e^x-1 / 2e^x+1

Accepted Answer

Hello, in this video, we are going to be calculating the derivative of the given function. The function given to us is F of X is equal to 3EX plus 2 divided by 3E the power of X minus 2. So, notice that our function is in the form of a rational function. In order to take the derivative of a rational function, we will go ahead and use the quotient rule. The quotient rule is defined as the following. If we want to take the derivative of a rational function, we will define the numerator and denominator as their own functions, F of X and G of X. The derivative of irrational function is defined as G of X multiplied by F X minus F of X multiplied by G of X, all divided by G of X squared. So, what we need to do is we need to define the numerator and denominator as their own functions. So, in our given rational function, the numerator is 3E to the power of X + 2. We will let F of X equal to 3E the power of X + 2. And for G of X, we will define G of X as the denominator, which will be 3E to the power of X minus 2. Next, we need to take the derivative of the numerator and denominator. By using the exponential rules on both the numerator and denominator. The derivative of the numerator will be defined as 3E to the power of X, and the derivative of the denominator will be defined as 3E to the power of X. Now that we have the 4 pieces of information we need, we can go ahead and plug this into the quotient rule in order to find the derivative of the overall FFX function. So, by using the quotient rule, we have G of X, which is 3E to the power of X minus 2 multiplied by the derivative of F of X, which is 3E to the power of X, minus the original F of X, which is 3E the power of X plus 2, multiplied by the derivative of G of X, which is 3E to the power of X. And this is all divided by the original denominator squared. 3 E to the power of X minus 2 squared. Now what we need to do is we need to go ahead and simplify the derivative. We can start by distributing 3E to the power of X to 3E to the power of X minus 2. We will do the same with 3E to the power of X distributed to 3E to the power of X plus 2. This will simplify the numerator to be 9 E to the power of 2 X. Minus 6E to the power of X, minus the quantity, 9 to the power of 2 X plus 6E to the power of X. And again, this is all divided by 3E to the power of X minus 2 squared. Next, what we can do is we can go ahead and distribute the -1 to 9E to the power of 2X plus 6E to the power of X. This will change the signs for those terms. So, what we are left with is 9E to the power of 2X minus 6E to the power of X, minus 9E to the power of 2 X, minus 6E to the power of X. What we will do is go ahead and combine like terms. We can combine both of the 9 E to the power of 2X and negative 9E to the power of 2X to cancel each other out. And -6E to the power of X minus 6E to the power of X will combine to leave us with negative 12 E to the power of X, divided by 3E to the power of X minus 2 squad. There is no further way to simplify the derivative. So this is going to be the derivative of the original F of X function. And this is going to be the solution to the problem. Furthermore, the overall solution to this problem will be a. 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) = 2e^x-1 / 2e^x+1

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