# What is (f(x+h) - f(x))/h when f(x) = x^2+9x-3?

Mar 5, 2018

$= {\lim}_{h \to 0} \frac{{\left(x + h\right)}^{2} + 9 \left(x + h\right) - 3 - \left({x}^{2} + 9 x - 3\right)}{h}$

$= {\lim}_{h \to 0} \frac{{x}^{2} + 2 x h + {h}^{2} + 9 x + 9 h - 3 - {x}^{2} - 9 x + 3}{h}$

$= {\lim}_{h \to 0} \frac{\cancel{{x}^{2}} + 2 x h + {h}^{2} + \cancel{9 x} + 9 h - \cancel{3} - \cancel{{x}^{2}} - \cancel{9 x} + \cancel{3}}{h}$

$= {\lim}_{h \to 0} \frac{2 x h + {h}^{2} + 9 h}{h}$

$= {\lim}_{h \to 0} \frac{h \left(2 x + h + 9\right)}{h}$

$= {\lim}_{h \to 0} \frac{\cancel{h} \left(2 x + h + 9\right)}{\cancel{h}}$

$= {\lim}_{h \to 0} 2 x + 0 + 9$

= 2x + 9