Evaluate the difference quotient for the given function. Simplify your answer.f(x) = −x3,    f(a + h) − f(a)h

Sure, let's evaluate the difference quotient for the function f(x) = -x^3.

The difference quotient formula is given by:

(f(a + h) - f(a)) / h

We can substitute the function f(x) = -x^3 into the difference quotient formula:

f(a + h) = -(a + h)^3 f(a) = -a^3

Substitute these into the difference quotient formula:

= [-(a + h)^3 - (-a^3)] / h = [-a^3 - 3a^2h - 3ah^2 - h^3 + a^3] / h = [-3a^2h - 3ah^2 - h^3] / h

Now, we can simplify by dividing each term in the numerator by h:

= -3a^2 - 3ah - h^2

So, the difference quotient for the function f(x) = -x^3 is -3a^2 - 3ah - h^2.