differentiate x*sin(x) — step-by-step solution

Let's perform the differentiation:

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

For

u1

x

and

v2

\sin (x)

, use the differentiation rule:

\frac{d}{d x} (u1 \cdot v2) = \frac{d}{d x} (u1) \cdot v2 + u1 \cdot \frac{d}{d x} (v2)

Let's perform the differentiation

u1

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

ⓘ The derivative of x is 1:

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

1

1

Let's perform the differentiation

v2

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

ⓘ The derivative of sin(x) is cos(x):

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

\cos (x)

\cos (x)

Let's evaluate the resulting derivative (

u1 \cdot v2

)':

1 \cdot \sin (x) + x \cdot \cos (x)

\sin (x) + x \cdot \cos (x)

Then you can resume the derivation:

ⅰ

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

\sin (x) + x \cdot \cos (x)

\sin (x) + x \cdot \cos (x)

▶Derivative:

x \cdot \cos (x) + \sin (x)