Limits, Continuity & Differentiability
Methods of Differentiation
Grade 12
Question:
<p>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 $g'(2)$ is equal to:</p>
<p>$\dfrac{1}{4}$</p>
<p>$\dfrac{1}{2e^{1/4}+5}$</p>
<p>$\dfrac{4}{e^{1/4}}$</p>
<p>$4+5\cdot 2^{4/5}$</p>
Step-by-Step Solution
Key Concept: General
<b>Inverse Function Derivative</b><br>
$g = f^{-1}$, so $g'(y_0)=1/f'(f^{-1}(y_0))$.<br>
Find $f^{-1}(2)$: solve $x^5+2e^{x/4}=2$. Test $x=0$: $0+2=2$ ✓. So $f(0)=2$, hence $g(2)=0$.<br>
$f'(x)=5x^4+\dfrac{e^{x/4}}{2}$. At $x=0$: $f'(0)=0+1/2=1/2$.<br>
$g'(2)=\dfrac{1}{f'(0)}=2$. So $g'(2)=2$, which is option (2). <b>Answer: 2</b><br>
<b>Key concept:</b> To find the inverse derivative, find the pre-image point first (test clean values), then apply $g'=1/f'$.<br>
<b>Trap:</b> Not recognising that $x=0$ maps to $f(0)=2$; testing $x=1$ or other values unnecessarily.
Correct Answer: 2