derivative of (sin(x)/(x^2)) — step-by-step solution

1) Let's perform the differentiation:

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

For

p1

\sin (x)

and

q2

x^{2}

, use the differentiation rule:

\frac{d}{d x} (\frac{p1}{q2}) = \frac{\frac{d}{d x} (p1) \cdot q2 - p1 \cdot \frac{d}{d x} (q2)}{{q2}^{2}}

2) Let's perform the differentiation

p1

\frac{d}{d x} (\sin (x))

ⓘ The derivative of sin(x) is cos(x):

\frac{d}{d x} (\sin (x))

\cos (x)

\cos (x)

3) Let's differentiate

q2

\frac{d}{d x} (x^{2})

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

2 x

4) We evaluate the resulting derivative (

\frac{p1}{q2}

)':

\frac{\cos (x) \cdot x^{2} - \sin (x) \cdot 2 x}{{(x^{2})}^{2}}

ⓘ What the terms have in common is factored out, so that the fraction can be simplified:

\frac{\cos (x) \cdot x^{2} - \sin (x) \cdot 2 x}{{(x^{2})}^{2}}

\frac{x \cdot (x \cdot \cos (x) - 2 \cdot \sin (x))}{{(x^{2})}^{2}}

■A power of a power multiplies the exponents, (a^n)^m = a^(n*m):

{(x^{2})}^{2}

x^{2 \cdot 2}

\frac{x \cdot (x \cdot \cos (x) - 2 \cdot \sin (x))}{x^{2 \cdot 2}}

ⅰ

2 \cdot 2

4

\frac{x \cdot (x \cdot \cos (x) - 2 \cdot \sin (x))}{x^{4}}

We simplify the fraction:

\frac{x}{x^{4}}

by the common factor:

x

to get the new quotient:

\frac{1}{x^{3}}

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

5) Then you can continue the differentiation:

ⅰ

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

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

▶Derivative:

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