21–30. Derivatives
a. Use limits to find the derivative ...

Question

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(x) = 4x²+1; a= 2,4

Accepted Answer

Hello. In this video, we are going to be determining the derivative of the function by using the definition of the derivative. The function given to us is G of X equal to 3 X2 minus 5. We want to go ahead and determine the derivative by using the definition of a derivative. Now, recall that the definition of a derivative states that if we take the limit as H approaches 0 of G of X plus H minus G of X divided by H, this limit will produce the derivative we are looking for. So, what do we need in order to start using this limit? The first thing we need is the G of X function. Now, that is already given to us in the problem, so we don't have to worry about finding that function. But what we do need to calculate is our G X plus H function. In order to do this, we will go ahead and take the quantity X + H and plug it into our given G of X function. That will give us G of X + H is equal to 3 multiplied by X + H quantity squared minus 5. Let's go ahead and simplify this function. We will expand the quantity X plus H2. That will leave us with 3 multiplied by X2 + 2 XH plus H2 minus 5. If we distribute 3 into the quantity, this will further simplify the function to be 3 X2 + 6 XH plus 3 H2 minus 5, and this is going to be our G of X plus H function. Now that we have both of the pieces we need in order to use the limit, let's go ahead and apply our two functions to the limit in order to solve for our derivative. That is going to give us the limit. As H approaches 0, Of G X plus H, which will be 3 X2 + 6 XH plus 3 H2 minus 5, subtract the G of X function, which is 3 X2 minus 5, and this will all be divided by H. So, let's go ahead and simplify the limit. The first thing we can do is distribute -1 into the quantity 3X2 minus 5. This will allow us to rewrite the quantity as negative 3X2 + 5. If we go ahead and combine like terms, we can eliminate the terms 3X2d and negative 3X2d. We can also eliminate -5 with 5. This will allow us to further simplify the limit to be the limit as H approaches 0 of 6 XH plus 3 H squared divided by H. Our goal is to eliminate the H in the denominator, but notice that the terms in the numerator have a multiple of H. What we can go ahead and do is factor out an H from the numerator of the limit. That will allow us to rewrite the limit to be the limit as H approaches 0 of H multiplied by the quantity, 6 X plus 3 H divided by H. We can eliminate the factors of H from both the numerator and denominator, and the simplest form of the limit is going to be the limit as H approaches 0 of 6 X plus 3 H. Finally, we are going to apply the limit by using direct substitution for the variable H. This will give us 6 X plus 3 multiplied by 0, which is going to simplify to be 6 X. What this means is that our derivative G of X is going to equal to 6 X by the definition of a derivative. And with that being said, the solution to this problem is going to be a. So I hope this video helps you in understanding on how to use the definition of a derivative, and I will go ahead and see you all in the next video.

21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 4x²+1; a= 2,4

Key Concepts

Derivative

Limit

Function Evaluation

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