Differentiate the following with respect to x. (i) sin2 x (ii) cos2

1.	Differentiate the following with respect to x. (i) sin2 x (ii) cos2 x (iii) cos3 x
Answer» For the following problems chain rule to be used: \(\frac{d}{dx}\)f(g(x)) = f'(g(x)).g'(x) \(\frac{d}{dx}\)[f(x)]ⁿ = n[f(x)]^{n - 1} x \(\frac{d}{dx}\)f(x) (i) Let y = sin² x = (sin x)² \(\frac{dy}{dx}\) = 2(sin x)^2-1 \(\frac{d}{dx}\)(sin x) = 2 sin x (cos x) = sin 2x (ii) y = cos² x = (cos x)² \(\frac{dy}{dx}\) = 2(cos x)^2-1 \(\frac{d}{dx}\)(cos x) = 2 cos x (-sin x) = -2 sin x cos x = -sin 2x (iii) y = cos³ x y = (cos x)³ \(\frac{dy}{dx}\) = 3(cos x)^{3 - 1} \(\frac{d}{dx}\)(cos x) = 3 cos² x (-sin x) = -3 cos² x sin x = -3 cos x (sin x cos x) [Multiply and divide by 2] = \(\frac{-3}{2}\) cos x (2 sin x cos x) = \(\frac{-3}{2}\) cos x sin 2x

Differentiate the following with respect to x. (i) sin2 x (ii) cos2 x (iii) cos3 x

Answer»

For the following problems chain rule to be used:

\(\frac{d}{dx}\)f(g(x)) = f'(g(x)).g'(x)

\(\frac{d}{dx}\)[f(x)]ⁿ = n[f(x)]^{n - 1} x \(\frac{d}{dx}\)f(x)

(i) Let y = sin² x = (sin x)²

\(\frac{dy}{dx}\) = 2(sin x)^2-1 \(\frac{d}{dx}\)(sin x)

= 2 sin x (cos x)

= sin 2x

(ii) y = cos² x = (cos x)²

\(\frac{dy}{dx}\) = 2(cos x)^2-1 \(\frac{d}{dx}\)(cos x)

= 2 cos x (-sin x)

= -2 sin x cos x

= -sin 2x

(iii) y = cos³ x

y = (cos x)³

\(\frac{dy}{dx}\) = 3(cos x)^{3 - 1} \(\frac{d}{dx}\)(cos x)

= 3 cos² x (-sin x)

= -3 cos² x sin x

= -3 cos x (sin x cos x) [Multiply and divide by 2]

= \(\frac{-3}{2}\) cos x (2 sin x cos x)

= \(\frac{-3}{2}\) cos x sin 2x