Calculus Examples

$f (x) = 4 \csc (2 x^{3} - 1)$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \csc (2 x^{3} - 1)$ with respect to $x$ is $4 \frac{d}{d x} [\csc (2 x^{3} - 1)]$ .

$4 \frac{d}{d x} [\csc (2 x^{3} - 1)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \csc (x)$ and $g (x) = 2 x^{3} - 1$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $2 x^{3} - 1$ .

$4 (\frac{d}{d u} [\csc (u)] \frac{d}{d x} [2 x^{3} - 1])$

Step 2.2

The derivative of $\csc (u)$ with respect to $u$ is $- \csc (u) \cot (u)$ .

$4 (- \csc (u) \cot (u) \frac{d}{d x} [2 x^{3} - 1])$

Step 2.3

Replace all occurrences of $u$ with $2 x^{3} - 1$ .

$4 (- \csc (2 x^{3} - 1) \cot (2 x^{3} - 1) \frac{d}{d x} [2 x^{3} - 1])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $4$ .

$- 4 (\csc (2 x^{3} - 1) \cot (2 x^{3} - 1) \frac{d}{d x} [2 x^{3} - 1])$

Step 3.2

By the Sum Rule, the derivative of $2 x^{3} - 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 1]$ .

$- 4 \csc (2 x^{3} - 1) \cot (2 x^{3} - 1) (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 1])$

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$- 4 \csc (2 x^{3} - 1) \cot (2 x^{3} - 1) (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 1])$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$- 4 \csc (2 x^{3} - 1) \cot (2 x^{3} - 1) (2 (3 x^{2}) + \frac{d}{d x} [- 1])$

Step 3.5

Multiply $3$ by $2$ .

$- 4 \csc (2 x^{3} - 1) \cot (2 x^{3} - 1) (6 x^{2} + \frac{d}{d x} [- 1])$

Step 3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$- 4 \csc (2 x^{3} - 1) \cot (2 x^{3} - 1) (6 x^{2} + 0)$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $6 x^{2}$ and $0$ .

$- 4 \csc (2 x^{3} - 1) \cot (2 x^{3} - 1) (6 x^{2})$

Step 3.7.2

Multiply $6$ by $- 4$ .

$- 24 \csc (2 x^{3} - 1) \cot (2 x^{3} - 1) x^{2}$

Step 3.7.3

Reorder the factors of $- 24 \csc (2 x^{3} - 1) \cot (2 x^{3} - 1) x^{2}$ .

$- 24 x^{2} \cot (2 x^{3} - 1) \csc (2 x^{3} - 1)$