Calculus Examples

$y = \frac{1}{\sin (20 x)}$

Step 1

Convert from $\frac{1}{\sin (20 x)}$ to $\csc (20 x)$ .

$\frac{d}{d x} [\csc (20 x)]$

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 $g (x) = 20 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $20 x$ .

$\frac{d}{d u} [\csc (u)] \frac{d}{d x} [20 x]$

Step 2.2

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

$- \csc (u) \cot (u) \frac{d}{d x} [20 x]$

Step 2.3

Replace all occurrences of $u$ with $20 x$ .

$- \csc (20 x) \cot (20 x) \frac{d}{d x} [20 x]$

Step 3

Differentiate.

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Step 3.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x$ with respect to $x$ is $20 \frac{d}{d x} [x]$ .

$- \csc (20 x) \cot (20 x) (20 \frac{d}{d x} [x])$

Step 3.2

Multiply $20$ by $- 1$ .

$- 20 \csc (20 x) \cot (20 x) \frac{d}{d x} [x]$

Step 3.3

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

$- 20 \csc (20 x) \cot (20 x) \cdot 1$

Step 3.4

Simplify the expression.

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Step 3.4.1

Multiply $- 20$ by $1$ .

$- 20 \csc (20 x) \cot (20 x)$

Step 3.4.2

Reorder the factors of $- 20 \csc (20 x) \cot (20 x)$ .

$- 20 \cot (20 x) \csc (20 x)$