# If  f(x)=x²-2x+3, how do you find f(a+h)-f(a)/h?

Oct 4, 2015

Substitute $a + h$ and $a$ for $x$ in the formula for $f \left(x\right)$ and simplify to find:

$\frac{f \left(a + h\right) - f \left(a\right)}{h} = 2 a - 2 + h$

#### Explanation:

$f \left(x\right) = {x}^{2} - 2 x + 3$

Then:

$\frac{f \left(a + h\right) - f \left(a\right)}{h}$

$= \frac{\left({\left(a + h\right)}^{2} - 2 \left(a + h\right) + 3\right) - \left({a}^{2} - 2 a + 3\right)}{h}$

$= \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{{a}^{2}}}} + 2 a h + {h}^{2} - \textcolor{b l u e}{\cancel{\textcolor{b l a c k}{2 a}}} - 2 h + \textcolor{g r e e n}{\cancel{\textcolor{b l a c k}{3}}} - \textcolor{red}{\cancel{\textcolor{b l a c k}{{a}^{2}}}} + \textcolor{b l u e}{\cancel{\textcolor{b l a c k}{2 a}}} - \textcolor{g r e e n}{\cancel{\textcolor{b l a c k}{3}}}}{h}$

$= 2 a - 2 + h$

So:

${\lim}_{h \to 0} \frac{f \left(a + h\right) - f \left(a\right)}{h} = {\lim}_{h \to 0} \left(2 a - 2 + h\right) = 2 a - 2$

This is the derivative of $f \left(x\right)$ at $x = a$