Question: Given the function f(x)=[h(p(x))]^2 and h(5)=1, h'(5)=2, h'(1)=3, p(1)=5, p'(1)=7, evaluate f'(1).
I tried solving what p(x) may be and figured the most trivial answer is p(x)=7x-2 (it meets the condition that p(1)=5=7(1)-2 and p'(1)=7=7(1)).
Next step is finding h(p(x)). I thought that I'd apply the chain rule here and figure out an equation to satisfy h'(5)=2 and h'(1)=3, but I can't figure it out.
How do I solve this?
Take the derivative: $$f'(x) = 2h(p(x))h'(p(x))p'(x)$$ Substitute $1$ in for $x$: $$f'(1) = 28$$