Assume f and g are differentiable functions with hx fgx Sup

Assume f and g are differentiable functions with h(x) = f(g(x)). Suppose the equation of the line tangent to the graph of g at the point (4,1) is y = 2x - 7 and the equation of the line tangent to the graph of f at (1,5) is y = 4x+ 1. Calculate h(4) and h\'(4). Determine an equation of the line tangent to the graph of h at the point on the graph where x=4.

Solution

h\'(x)=f\'(g(x)*g\'(x) h\'(4)=f\'(g(4))*g\'(4) =f\'(1)*g\'(4) =2*4=8 h(4)=f(g(4)=f(1)=5 eq y-5=8(x-4) y=8x+27