Calculate: d d x sin 2 ⁡ x cos 3 ⁡ x

Question

Calculate:      d          d      x            sin          2        ⁡  x      cos          3        ⁡  x

apopihvj · Accepted Answer

Use Product Rule to find the derivative of

\sin^{2} x \cos^{3} x

. The product rule states that

(f g)' = f' g + f g'

.

(\frac{d}{d x} \sin^{2} x) \cos^{3} x + \sin^{2} x (\frac{d}{d x} \cos^{3} x)

Use Chain Rule on

\frac{d}{d x} \sin^{2} x

. Let

u = \sin x

. Use Power Rule:

\frac{d}{d u} u^{n} = n u^{n - 1}

.

2 \sin x (\frac{d}{d x} \sin x) \cos^{3} x + \sin^{2} x (\frac{d}{d x} \cos^{3} x)

Use Trigonometric Differentiation: the derivative of

\sin x

is

\cos x

.

2 \sin x \cos^{4} x + \sin^{2} x (\frac{d}{d x} \cos^{3} x)

Use Chain Rule on

\frac{d}{d x} \cos^{3} x

. Let

u = \cos x

. Use Power Rule:

\frac{d}{d u} u^{n} = n u^{n - 1}

.

2 \sin x \cos^{4} + \sin^{2} x \times 3 \cos^{2} x (\frac{d}{d x} \cos x)

Use Trigonometric Differentiation: the derivative of

\cos x

is -

\sin x

2 \sin x \cos^{4} x - 3 \sin^{3} x \cos^{2} x

Calculate: (d)/(dx) sin(x)^2 cos(x)^3 xdx