Question

Using the definition of the derivative, find f'(x). Then find f'(-1), f'(0), and f'(4) when the derivative exists. fx)=6x³ +3. To find the derivative, complete the limit as h approaches 0 for f(x +h)-f(x)/h

Answer 1

To find the derivative of the function f(x) = 6x³ + 3, use the definition of the derivative. Plug in the values of x to find f'(-1), f'(0), and f'(4) when the derivative exists.

Step-by-step explanation:

To find the derivative of the function f(x) = 6x³ + 3, we can use the definition of the derivative. The derivative is found by taking the limit as h approaches 0 of [f(x + h) - f(x)] / h. Plugging in the function, we get [6(x + h)³ + 3 - 6x³ - 3] / h. Expanding and simplifying, we have [6x³ + 18x²h + 18xh² + 6h³ + 3 - 6x³ - 3] / h, which simplifies further to 18x² + 18xh + 6h² / h. Canceling out h, we get the derivative f'(x) = 18x² + 18xh + 6h².

To find f'(-1), f'(0), and f'(4), we can plug in the respective values of x into the derivative function:

f'(-1) = 18(-1)² + 18(-1)h + 6h² = 18 - 18h + 6h²

f'(0) = 18(0)² + 18(0)h + 6h² = 6h²

f'(4) = 18(4)² + 18(4)h + 6h² = 288 + 72h + 6h²