derivative of x(1-cos(x)) — step-by-step solution

1) Let's differentiate:

\frac{d}{d x} (x \cdot (1 - \cos (x)))

We differentiate

u1

x

and

v2

1 - \cos (x)

using:

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

2) Let's perform the differentiation

u1

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

ⓘ The derivative of x is 1:

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

1

1

3) Let's perform the differentiation

v2

\frac{d}{d x} (1 - \cos (x))

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (1 - \cos (x))

0 + \frac{d}{d x} (- \cos (x))

0 + \frac{d}{d x} (- \cos (x))

ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (- \cos (x))

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

■The derivative of cos(x) is -sin(x):

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

\sin (x)

\sin (x)

4) Let's evaluate the resulting derivative (

u1 \cdot v2

)':

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

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

5) Then you can continue the differentiation:

ⅰ

\frac{d}{d x} (x \cdot (1 - \cos (x)))

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

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

▶Derivative:

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