## Calculus 10th Edition

$8x +4 \Delta x$, $\Delta x\ne0$
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: $8x +4 \Delta x$ $\Delta x \ne0$