Question

The derivative of the function

f(x) = cos^–1$\left\{ \frac{1}{\sqrt{13}}(2\cos x - 3\sin x) \right\}$+ sin^–1

$\left\{ \frac{1}{\sqrt{13}}(2\sin x + 3\cos x) \right\}$with respect to $\sqrt{1 + x^{2}}$ is –

• A. 2x
• B. $2\sqrt{1 + x^{2}}$
• C. $\frac{2}{x}\sqrt{1 + x^{2}}$
• D. $\frac{2x}{\sqrt{1 + x^{2}}}$

Answer 1

Answer: (C)

$\frac{2}{x}\sqrt{1 + x^{2}}$

Answer 2

With cos θ = $\frac{2}{\sqrt{13}}$and sin θ =$\frac{3}{\sqrt{13}}$,

f(x) = cos^–1(cos θ cos x – sin θ sin x) + sin^–1

(cos θ sin x + sin θ cos x)

= cos^–1(cos (θ + x) + sin^–1(sin (θ + x))

= 2x + 2θ

By chain rule, the required derivative

= f '(x) $\frac{1}{\frac{d}{dx}\left( \sqrt{1 + x^{2}} \right)}$ = $\frac{2}{x}\sqrt{1 + x^{2}}$