i)Find h'(2) given that h(x)=f(g(x)), f(u)=u^2-1, g(2)=3, g'(2)=-1
ii) let y=f(x^2+3x-5). find dy/dx when x=1, given that f'(-1)=2
We have the function $\displaystyle f(u)=u^2-1$ if we set $\displaystyle u=g(x)$ we get the function $\displaystyle h(x)=f(g(x))=(g(x))^2-1$ (notice how I substituted $\displaystyle u=g(x)$). We use the chain rule to differentiate:
$\displaystyle h'(x)=f'(g(x))g'(x)$.
$\displaystyle =2g(x)g'(x)$.
Do you understand how I got $\displaystyle h'(x)=2g(x)g'(x)$?
If so, then we have
$\displaystyle h'(2)=2g(2)g'(2)$
$\displaystyle h'(2)=2(3)(-1)=-6$