Limits, Continuity & Differentiability
Methods of Differentiation
Grade None
Question:
<p>Let $f(x)=x^3+3x+1$ and $g$ be its inverse. Then $g'(5)$:</p>
<p>$\dfrac{1}{5}$</p>
<p>$\dfrac{1}{6}$</p>
<p>$\dfrac{1}{3}$</p>
<p>$\dfrac{1}{7}$</p>
Step-by-Step Solution
Key Concept: General
<b>Inverse at a Specific Value</b><br>
Solve $f(a)=5$: $a^3+3a+1=5\Rightarrow a^3+3a-4=0\Rightarrow(a-1)(a^2+a+4)=0\Rightarrow a=1$ (real root).<br>
$g'(5)=1/f'(1)=1/(3\cdot1+3)=1/6$. Option (2).<br>
<b>Answer: 2 (= 1/6)</b><br>
<b>Key concept:</b> Always find the pre-image first by factoring $f(a)=y_0$, then evaluate $1/f'(a)$.<br>
<b>Trap:</b> Solving $f'(x)=5$ instead of $f(x)=5$.
Correct Answer: 2