find the difference quotient and simplify. f(x)=x^3 - 1 the difference quotient of f(x) is

Explanation:

Step1: Recall the difference - quotient formula

The difference - quotient formula is $\frac{f(x + h)-f(x)}{h}$, where $f(x)=x^{3}-1$ and $f(x + h)=(x + h)^{3}-1$.

Step2: Expand $(x + h)^{3}$

Using the formula $(a + b)^{3}=a^{3}+3a^{2}b + 3ab^{2}+b^{3}$, we have $(x + h)^{3}=x^{3}+3x^{2}h + 3xh^{2}+h^{3}$. So $f(x + h)=x^{3}+3x^{2}h + 3xh^{2}+h^{3}-1$.

Step3: Substitute $f(x + h)$ and $f(x)$ into the difference - quotient formula

$\frac{f(x + h)-f(x)}{h}=\frac{(x^{3}+3x^{2}h + 3xh^{2}+h^{3}-1)-(x^{3}-1)}{h}$.

Step4: Simplify the numerator

$(x^{3}+3x^{2}h + 3xh^{2}+h^{3}-1)-(x^{3}-1)=x^{3}+3x^{2}h + 3xh^{2}+h^{3}-1 - x^{3}+1=3x^{2}h + 3xh^{2}+h^{3}$.

Step5: Simplify the fraction

$\frac{3x^{2}h + 3xh^{2}+h^{3}}{h}=\frac{h(3x^{2}+3xh + h^{2})}{h}=3x^{2}+3xh + h^{2}$.

Answer:

$3x^{2}+3xh + h^{2}$