Applications of Derivatives
Derivative of Inverse Function
nta_pyq_2024_apr
Grade 12
Question:
Let $f(x)=x^5+2e^{x/4}$ for all $x\in\mathbb{R}$. Consider a function $g(x)$ such that $(g\circ f)(x)=x$ for all $x\in\mathbb{R}$. Then the value of $8g'(2)$ is:
Step-by-Step Solution
Key Concept: $g$ is the inverse of $f$. $g'(f(x))\cdot f'(x)=1$. Find $x$ such that $f(x)=2$: at $x=0$, $f(0)=0+2e^0=2$. $g'(2)=1/f'(0)$.
$f(0)=2$, $f'(0)=1/2$. $g'(2)=1/(1/2)=2$. $8g'(2)=16$.
Correct Answer: 4