Differential Calculus
Iterated composite function derivative
MJMT_Full_Test_09
Grade 12
Question:
Let $f,g,h:\mathbb{R}\to\mathbb{R}$ be differentiable with $f(x)=x^5+x^3+3x+7$, $g(f(x))=x$ and $h(g(g(x)))=x$. Value of $h'(-1)$ is
Step-by-Step Solution
Key Concept: $g=f^{-1}$ (so $g'=1/f'(g(x))$). $h=g^{-1}\circ g^{-1}=f\circ f$... actually $h(g(g(x)))=x\Rightarrow h=(g\circ g)^{-1}=f\circ f$. $h'(-1)=f'(f(-1))\cdot f'(-1)$.
$h'(-1)=1045$.
Correct Answer: 1045