Question

Let f(x)=4-x^2,find ( f(x+h)-f(x) ) / h and simplify

Answer 1

If $f(x)=4-x^2$, then

$f(x+h)=4-(x+h)^2=4-(x^2+2xh+h^2)=4-x^2-2xh-h^2$

So

$\dfrac{f(x+h)-f(x)}h=\dfrac{(4-x^2-2xh-h^2)-(4-x^2)}h=-\dfrac{2xh+h^2}h$

Then for $h\neq0$, we can cancel common terms:

$\dfrac{f(x+h)-f(x)}h=-2x+h$