Answer
$8x +4 \Delta x$,
$\Delta x\ne0$
Work Step by Step
Given: $f(x) = 4x^2$
Evaluate:
$\frac{f(x +\Delta x) - f(x)}{\Delta x}$
$\frac{4(x +\Delta x)^2 - 4x^2}{\Delta x}$
Note: $\Delta x \ne0$
Expand:
$\frac{4(x^2 + 2x\Delta x + \Delta x^2) - 4x^2}{\Delta x}$
Distribute:
$\frac{4x^2 + 8x\Delta x + 4\Delta x^2 - 4x^2}{\Delta x}$
Add like terms: $\frac{ 8x\Delta x + 4\Delta x^2 }{\Delta x}$
Simplify: $8x +4 \Delta x$
$\Delta x \ne0$
