f(x)=5x/(x2+9); f(x+h)=5(x+h)/((x+h)2+9)=5(x+h)/(x2+2hx+h2+9).
f(x+h)-f(x)=5(x+h)/(x2+2hx+h2+9)-5x/(x2+9)=
(5(x+h)(x2+9)-5x(x2+2hx+h2+9))/[(x2+9)(x2+2hx+h2+9)]=
(5x3+45x+5hx2+45h-5x3-10hx2-45x)/(x4+2hx3+x2h2+18x2+18hx+9h2+81)=
(45h-5hx2)/((x4+18x2+81)+2hx3+x2h2+9h2).
Divide by h:
(45-10x2)/((x2+9)2+2hx3+x2h2+9h2).
In the denominator we can let h→0, so we have the gradient (derivative):
(45-5x2)/(x2+9)2, which is the limit as h→0.
If f(x)=10/x2, then f(x+h)=10/(x+h)2,
so f(x+h)-f(x)=10(1/(x+h)2-1/x2)=10(x2-(x+h)2)/(x2(x+h)2)=10(-2xh-h2)/(x4+2x3h+x2h2).
Divide by h:
10(-2x-h)/(x4+2x3h+x2h2).
When h→0:
-20x/x4=-20/x3.