Find the derivative of the following functions (it is to be under…

Question

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\sin^{\text{n}}\text{x}$

✓

Answer

Let $\text{y}=\sin^{\text{n}}\text{x}.$Accordingly, for n = 1, y = sin x.$\therefore\frac{\text{dy}}{\text{dx}}=\cos\text{x},\text{ i.e.,}\frac{\text{d}}{\text{dx}}\sin\text{x}=\cos\text{x}$
For n = 2, y = sin^2 x. $\therefore\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(\sin\text{x}\sin\text{x})$
$=(\sin\text{x})'\sin\text{x}+\sin\text{x}(\sin\text{x})'\ [\text{By Leibnitz product rule]}$
$=\cos\text{x}\sin\text{x}+\sin\text{x}\cos\text{x}$
$=2\sin\text{x}\cos\text{x}\ ...(1)$For n = 3, y = sin^3 x.
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(\sin\text{x}\sin^2\text{x})$
$=(\sin\text{x})'\sin^2\text{x}+\sin\text{x}(\sin^2\text{x})'\ [\text{By Leibnitz product rule]}$$=\cos\text{x}\sin^2\text{x}+\sin(2\sin\text{x}\cos\text{x})\ [\text{Using (1)]}$
$=\cos\text{x}\sin^2\text{x}+2\sin^2\text{x}\cos\text{x}$
$=3\sin^2\text{x}\cos\text{x}$ We assert that $\frac{\text{d}}{\text{dx}}(\sin^{\text{n}}\text{x})=\text{n}\sin^{(\text{n}-1)}\text{x}\cos\text{x}$Let our assertion be true for n = k. i. e., $\frac{\text{d}}{\text{dx}}(\sin^{\text{k}}\text{x})=\text{k}\sin^{(\text{k}-1)}\text{x}\cos\text{x}\ ...(2)$
$\text{Consider}\frac{\text{d}}{\text{dx}}(\sin^{\text{k}+1}\text{x})=\frac{\text{d}}{\text{dx}}(\sin\text{x}\sin^{\text{k}}\text{x})$
$=(\sin\text{x})'\sin^{\text{k}}\text{x}+\sin\text{x}(\sin^{\text{k}}\text{x})'\ [\text{By Leibnitz product rule]}$
$=\cos\text{x}\sin^{\text{k}}\text{x}+\sin\text{x}(\text{k}\sin^{(\text{k}-1)}\text{x}\cos\text{x})\ [\text{Using (2)]}$
$=\cos\text{x}\sin^{\text{k}}\text{x}+\text{k}\sin^{\text{k}}\text{x}\cos\text{x}$
$=(\text{k+1})\sin^{\text{k}}\text{x}\cos\text{x}$ Thus, our assertion is true for n = k + 1. Hence, by mathematical induction, $\frac{\text{d}}{\text{dx}}(\sin^{\text{n}}\text{x})=\text{n}\sin^{(\text{n}-1)}\text{x}\cos\text{x}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 2 marks)" from LIMITS AND DERIVATIVES→LIMITS AND DERIVATIVES — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

LIMITS AND DERIVATIVES — MATHS STD 11 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions