If #F(x)=f(xf(xf(x)))# where f(1)=2, f(2)=3, f'(1)=4, f'(2)=5, and f'(3)=6, how do you find F'(1)?

1 Answer
Nov 21, 2016

#F'(1)=84#

Explanation:

We will differentiate #F# and wade our way slowly and methodically through finding the derivative of the right. First with the outermost function and chain rule:

#F(x)=f(color(blue)(xf(xf(x))))#

#F'(x)=f'(color(blue)(xf(xf(x))))(d/dxcolor(blue)(xf(xf(x))))#

Now applying the product rule:

#F'(x)=f'(xf(xf(x)))[f(xf(x))+x(d/dxf(xf(x)))]#

Reapplying the chain rule:

#F'(x)=f'(xf(xf(x)))[f(xf(x))+x(f'(xf(x))+(d/dx xf(x)))]#

#F'(x)=f'(xf(xf(x)))[f(xf(x))+xf'(xf(x))+x(d/dx xf(x))]#

Product rule once more:

#F'(x)=f'(xf(xf(x)))[f(xf(x))+xf'(xf(x))+x(f(x)+xf'(x))]#

We could simplify this a little more, but I'm dubious it would help. Evaluating at #x=1#:

#F'(1)=f'(f(f(1)))[f(f(1))+f'(f(1))+1(f(1)+f'(1))]#

#F'(1)=f'(f(2))[f(2)+f'(2)+2+4]#

#F'(1)=f'(3)(3+5+2+4)#

#F'(1)=6(14)#

#F'(1)=84#