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(t) = 3t⁴; a= -2, 2

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with derivatives. This problem says to determine the derivative of the function F of X, which is equal to 5 multiplied by X-rays to the 4th, plus 2 using limits. We're given 4 possible choices as our answers. For choice A, we have F of X is equal to 5 x 2. For choice B, we have F of X, which is equal to 20 x 2. For C, we have F of X, which is equal to 5 x cubed. And for choice D we have F of X, which is equal to 20 x cubed. Now, we're asked to find the derivative of this function using limits. So we're going to be looking for F of X. And now recall your definition of a derivative in terms of limits. So this is going to be equal to the limit. As H approaches 0, of the quantity of F of X plus H. Minus F of X. In quantity, divided by H. So now we'll use our function that we're given in the problem, so this is going to be equal to the limit as H approaches 0, and for F of X plus H in our function F of X, we'll replace X with the quantity of X plus H. So we're going to have the quantity of 5 multiplied by the quantity of X plus H in quantity raised to the 4th power, plus 2, then minus the quantity, which is F of X. And that is 5 multiplied by X-rays to the 4th plus 2, in quantity, and all of that is divided by H. So now we just need to um simplify our numerator in this quotient. So this is going to be equal to the limit, as H approaches 0. Of the quantity of 5, and here we'll go ahead and multiply out the quantity of X plus H in quantity rates to the 4th power. So we'll have 5 multiplying the quantity of X rates to the 4th. Plus 4 multiplied by X cubed, multiplied by H. Plus 6 X 2 multiplied by H2. Plus 4 X multiplied by H cubed. Plus H rates to the 4th power in quantity. Then we'll have plus 2, then we'll distribute the minus sign in the next two terms, so we'll have minus 5, X-rays to the 4th. Minus 2, and all of that is divided by H. So now we'll distribute the 5 in the numerator. This is going to be equal to the limit. As H approaches 0, of the quantity of 5 multiplied by X-rays to the 4th, plus 20 X cubed. Multiplied by H plus 30 X2, multiplied by H2. Plus 20 X multiplied by H cubed. Plus 5 multiplied by 8 rates to the 4th. Then the next term, which is a + 2, can be added to the -2, and so those will add to 0, and then we're just left with minus 5, X-rays to the 4th, and all of that is still divided by H. Now, the 5 multiplied by X-rays to the 4th and the minus 5. Multiplied by X-rays to the 4th, we'll add to 0, and the remaining terms in the numerator all have H in them, so I can cancel out one factor of H with the H in the denominator. So this is going to be equal to the limit, as H approaches 0. Of the quantity of 2 execute. Plus 30 X squared multiplied by H. Plus 20 X multiplied by H2d, plus 5 multiplied by H cubed. So at this point, I can use direct substitution to evaluate our limit. And so we'll set H equal to 0, so this is going to be equal to. 20 ex cubed. What 30 X squared multiplied by 0, plus 20 X multiplied by 0 squared. Plus 5 multiplied by 0 Q. And evaluating this, this is going to be equal to. 20 execute. And so, this is the answer that we were looking for. And if we compare our answer to our answer choices, the correct answer choice corresponds to answer D. So I hope that this explanation has been useful, and I'll see you in the next video.

21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(t) = 3t⁴; a= -2, 2

Key Concepts

Derivative

Limit

Power Rule

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