Finding Derivative Functions and Values

Question

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

f(x) = 4 – x²; f′(−3), f′(0), f′(1)

Accepted Answer

Welcome back, everyone, to another video. Given the function F of X equals 18 minus X2din F of X using the limit definition of the derivative, then calculate the value of F 3. So, first of all, we're going to identify F of X using the limit definition. It says that we're evaluating limit as H approaches 0 of FX plus H minus F of X divided by H. Now that we have the required expression, we're going to evaluate each term, which gives us limit as H 4 to 0. F X plus H is going to be 18 minus X + H2, right? So this is going to be our first term F of X plus H, and then we are subtracting F of X, which is given 18 minus X2d. All of that is divided by H. And now what we can do from here is basically rewrite our limit as limit as H approaches 0 of 18 minus let's square x plus H. This gives us X2 + 2 XH plus H2, and then we're subtracting 18 and adding X2d by applying the distributive property. All of that is divided by H. What we want to do from here is combine the like terms. So we get limit as H approaches 0 of 18 minus X2 minus 2 XH minus H2 minus 18 + X2. So far we have applied the distributive property once again. We have to divide it by H, and we can start combining the like terms, which gives us limit as H approaches 0. 1st of all, we can cancel out 18 and negative 18, negative X2d and X2, and then we end up with two terms. We can also factor out H. Or negative H. And then er we have 2 Xs. Plus H which comes from negative H2. This is divided by H, and we can now see that we can cancel out H, which is the common factor within our fraction. Therefore, we got limit as H approaches 0. Of Negative. 2 X plus H. and because H approaches 0, we can substitute H equals 0, and overall we get -2 X. Well done, so we have concluded that F of X is -2 X, and now FF3 is going to be equal to. -2 multiplied by 3, right? Basically we have to substitute X equals 3 into the derivative, so we get -2 multiplied by 3, which is -6. So our next answer is -6. That said, we have solved the problem. The derivative is -2 X and the value of the derivative at 3 is -6. Thank you for watching.

Finding Derivative Functions and Values

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

f(x) = 4 – x²; f′(−3), f′(0), f′(1)

Key Concepts

Definition of Derivative

Power Rule for Derivatives

Evaluating Derivatives at Specific Points

Watch next