Calculus Examples

$y = 2 \csc (1 - 3 x)$

Step 1

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

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

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 $g (x) = 1 - 3 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $1 - 3 x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $1 - 3 x$ .

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

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $2$ .

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

Step 3.2

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

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

Step 3.3

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

$- 2 \csc (1 - 3 x) \cot (1 - 3 x) (0 + \frac{d}{d x} [- 3 x])$

Step 3.4

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

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

Step 3.5

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

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

Step 3.6

Multiply $- 3$ by $- 2$ .

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

Step 3.7

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

$6 \csc (1 - 3 x) \cot (1 - 3 x) \cdot 1$

Step 3.8

Simplify the expression.

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Step 3.8.1

Multiply $6$ by $1$ .

$6 \csc (1 - 3 x) \cot (1 - 3 x)$

Step 3.8.2

Reorder the factors of $6 \csc (1 - 3 x) \cot (1 - 3 x)$ .

$6 \cot (1 - 3 x) \csc (1 - 3 x)$