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 (1,6) is y = - 3x + 9 and the equation of the line tangent to the graph of f at (6,9) is y= - 2x + 21. Calculate h(1) and h\'(1). Determine an equation of the line tangent to the graph of h at the point on the graph where x=1.

Solution

h(1)=f(g(1))=f(6)=9 h\'(1)=f\'(g(1)*g\'(1)=-2*-3=6 y-9=6(x-1) y=6x+3