Question

Suppose that f(x h)−f(x)=−1hx²−6hx−2h²x−7h² 6h³. find f′(x).

Answer 1

To find the derivative of the function f(x), we need to calculate the limit of the difference quotient as h approaches 0. The derivative of f(x) is -x^2 - 6x.

Step-by-step explanation:

To find the derivative of the function f(x), we need to calculate the limit of the difference quotient as h approaches 0.

Using the formula for the difference quotient and simplifying, we can write:

f'(x) = lim(h->0) [-1x^2 - 6x - 2hx - 7h^2]/(6h^3)

Next, we can factor out an h from the numerator and cancel out the h in the denominator:

f'(x) = lim(h->0) [-1x^2 - 6x - 2hx - 7h^2]/(6h^3) = lim(h->0) [-1x^2 - 6x - 2x - 7h]/(6h^2)

Finally, plugging in h = 0, we get f'(x) = -x^2 - 6x.