Applications of Derivatives
Derivative from First Principles — Limit Evaluation
nta_pyq_2024_jan
Grade 12
Question:
Suppose $f(x)=\dfrac{(2^x+2^{-x})\tan x\sqrt{\tan^{-1}(x^2-x+1)}}{(7x^2+3x+1)^3}$. Then the value of $f'(0)$ is equal to
$\pi$
$0$
$\sqrt{\pi}$
$\dfrac{\pi}{2}$
Step-by-Step Solution
Key Concept: Use $f'(0)=\lim_{h\to0}\frac{f(h)-f(0)}{h}$. Note $f(0)=0$ since $\tan(0)=0$. Expand each factor to leading order as $h\to0$ to evaluate the limit.
$f'(0)=\lim_{h\to0}\frac{f(h)}{h}$. As $h\to0$: numerator $\sim2h\cdot\sqrt{\pi/4}=h\sqrt{\pi}$, denominator $\to h$. $f'(0)=\sqrt{\pi}$.
Correct Answer: 3