derivative of x^3csc(x) — step-by-step solution

1) Let's perform the differentiation:

\frac{d}{d x} (x^{3} \cdot \csc (x))

For

u1

x^{3}

and

v2

\csc (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)

2) Let's differentiate

u1

\frac{d}{d x} (x^{3})

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{3})

3 x^{2}

3 x^{2}

3) Let's perform the differentiation

v2

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

ⓘ The derivative of csc(x) is -csc(x)*cot(x):

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

- \csc (x) \cdot \cot (x)

- \csc (x) \cdot \cot (x)

4) Let's evaluate the resulting derivative (

u1 \cdot v2

)':

3 x^{2} \cdot \csc (x) + x^{3} \cdot (- \csc (x) \cdot \cot (x))

ⓘ The minus sign is placed in front of the product:

x^{3} \cdot (- \csc (x))

- x^{3} \cdot \csc (x)

3 x^{2} \cdot \csc (x) - x^{3} \cdot \csc (x) \cdot \cot (x)

5) You can then resume the differentiation:

ⅰ

\frac{d}{d x} (x^{3} \cdot \csc (x))

3 x^{2} \cdot \csc (x) - x^{3} \cdot \csc (x) \cdot \cot (x)

3 x^{2} \cdot \csc (x) - x^{3} \cdot \csc (x) \cdot \cot (x)

3 x^{2} \cdot \csc (x) - x^{3} \cdot \cot (x) \cdot \csc (x)

x^{2} \cdot \csc (x) \cdot (3 - x \cdot \cot (x))

▶Derivative:

x^{2} \cdot \csc (x) \cdot (3 - x \cdot \cot (x))