Calculus Examples

$\frac{1}{2 \sin (2 x)}$

Step 1

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

$\frac{1}{2} \frac{d}{d x} [\frac{1}{\sin (2 x)}]$

Step 2

Convert from $\frac{1}{\sin (2 x)}$ to $\csc (2 x)$ .

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

Step 3

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 $g (x) = 2 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 4

Differentiate.

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Step 4.1

Combine $\frac{1}{2}$ and $\csc (2 x)$ .

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

Step 4.2

Combine $\cot (2 x)$ and $\frac{\csc (2 x)}{2}$ .

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

Step 4.3

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

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

Step 4.4

Simplify terms.

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Step 4.4.1

Multiply $2$ by $- 1$ .

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

Step 4.4.2

Combine $- 2$ and $\frac{\cot (2 x) \csc (2 x)}{2}$ .

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

Step 4.4.3

Cancel the common factor of $- 2$ and $2$ .

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Step 4.4.3.1

Factor $2$ out of $- 2 \cot (2 x) \csc (2 x)$ .

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

Step 4.4.3.2

Cancel the common factors.

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Step 4.4.3.2.1

Factor $2$ out of $2$ .

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

Step 4.4.3.2.2

Cancel the common factor.

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

Step 4.4.3.2.3

Rewrite the expression.

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

Step 4.4.3.2.4

Divide $- \cot (2 x) \csc (2 x)$ by $1$ .

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

Step 4.5

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

$- \cot (2 x) \csc (2 x) \cdot 1$

Step 4.6

Multiply $- 1$ by $1$ .

$- \cot (2 x) \csc (2 x)$