# Evaluate the difference quotient for the given function. Simplify your answer. f(x)=X^3, f(a+h)-frac{a}{h}

Evaluate the difference quotient for the given function. Simplify your answer. $f\left(x\right)={X}^{3},f\left(a+h\right)-\frac{a}{h}$
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Tuthornt
$\frac{f\left(a+h\right)-f\left(a\right)}{h}$
It is given that $f\left(x\right)={x}^{3}$
$=\frac{\left(a+h{\right)}^{3}-{a}^{3}}{h}$
Recall that: ${A}^{3}-{B}^{3}=\left(A-B\right)\left({A}^{2}+AB+{B}^{2}\right)$
$=\frac{\left(a+h-a\right)\left(a+h{\right)}^{2}+a\left(a+h\right)+{a}^{2}}{h}$
$=\frac{\left(h\right)\left[\left({a}^{2}+2ah+{h}^{2}\right)+\left({a}^{2}+ah\right)+{a}^{2}\right]}{h}$
Cancel h from both the numerator and the denominator
$=3{a}^{2}+3ah+{h}^{2}$
Result
$3{a}^{2}+3ah+{h}^{2}$