Calculus Examples

$\frac{\tan (6 x)}{\sin (2 x)}$

Step 1

Rewrite $\tan (6 x)$ in terms of sines and cosines.

$\frac{d}{d x} [\frac{\frac{\sin (6 x)}{\cos (6 x)}}{\sin (2 x)}]$

Step 2

Rewrite $\frac{\frac{\sin (6 x)}{\cos (6 x)}}{\sin (2 x)}$ as a product.

$\frac{d}{d x} [\frac{\sin (6 x)}{\cos (6 x)} \cdot \frac{1}{\sin (2 x)}]$

Step 3

Simplify.

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Step 3.1

Convert from $\frac{\sin (6 x)}{\cos (6 x)}$ to $\tan (6 x)$ .

$\frac{d}{d x} [\tan (6 x) \frac{1}{\sin (2 x)}]$

Step 3.2

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

$\frac{d}{d x} [\tan (6 x) \csc (2 x)]$

Step 4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \tan (6 x)$ and $g (x) = \csc (2 x)$ .

$\tan (6 x) \frac{d}{d x} [\csc (2 x)] + \csc (2 x) \frac{d}{d x} [\tan (6 x)]$

Step 5

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 5.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\tan (6 x) (\frac{d}{d u_{1}} [\csc (u_{1})] \frac{d}{d x} [2 x]) + \csc (2 x) \frac{d}{d x} [\tan (6 x)]$

Step 5.2

The derivative of $\csc (u_{1})$ with respect to $u_{1}$ is $- \csc (u_{1}) \cot (u_{1})$ .

$\tan (6 x) (- \csc (u_{1}) \cot (u_{1}) \frac{d}{d x} [2 x]) + \csc (2 x) \frac{d}{d x} [\tan (6 x)]$

Step 5.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$\tan (6 x) (- \csc (2 x) \cot (2 x) \frac{d}{d x} [2 x]) + \csc (2 x) \frac{d}{d x} [\tan (6 x)]$

Step 6

Differentiate.

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Step 6.1

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

$\tan (6 x) (- \csc (2 x) \cot (2 x) (2 \frac{d}{d x} [x])) + \csc (2 x) \frac{d}{d x} [\tan (6 x)]$

Step 6.2

Multiply $2$ by $- 1$ .

$\tan (6 x) (- 2 \csc (2 x) \cot (2 x) \frac{d}{d x} [x]) + \csc (2 x) \frac{d}{d x} [\tan (6 x)]$

Step 6.3

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

$\tan (6 x) (- 2 \csc (2 x) \cot (2 x) \cdot 1) + \csc (2 x) \frac{d}{d x} [\tan (6 x)]$

Step 6.4

Multiply $- 2$ by $1$ .

$\tan (6 x) (- 2 \csc (2 x) \cot (2 x)) + \csc (2 x) \frac{d}{d x} [\tan (6 x)]$

Step 7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 6 x$ .

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Step 7.1

To apply the Chain Rule, set $u_{2}$ as $6 x$ .

$\tan (6 x) (- 2 \csc (2 x) \cot (2 x)) + \csc (2 x) (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [6 x])$

Step 7.2

The derivative of $\tan (u_{2})$ with respect to $u_{2}$ is $\sec^{2} (u_{2})$ .

$\tan (6 x) (- 2 \csc (2 x) \cot (2 x)) + \csc (2 x) (\sec^{2} (u_{2}) \frac{d}{d x} [6 x])$

Step 7.3

Replace all occurrences of $u_{2}$ with $6 x$ .

$\tan (6 x) (- 2 \csc (2 x) \cot (2 x)) + \csc (2 x) (\sec^{2} (6 x) \frac{d}{d x} [6 x])$

Step 8

Differentiate.

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Step 8.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\tan (6 x) (- 2 \csc (2 x) \cot (2 x)) + \csc (2 x) \sec^{2} (6 x) (6 \frac{d}{d x} [x])$

Step 8.2

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

$\tan (6 x) (- 2 \csc (2 x) \cot (2 x)) + \csc (2 x) \sec^{2} (6 x) (6 \cdot 1)$

Step 8.3

Simplify the expression.

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Step 8.3.1

Multiply $6$ by $1$ .

$\tan (6 x) (- 2 \csc (2 x) \cot (2 x)) + \csc (2 x) \sec^{2} (6 x) \cdot 6$

Step 8.3.2

Move $6$ to the left of $\csc (2 x) \sec^{2} (6 x)$ .

$\tan (6 x) (- 2 \csc (2 x) \cot (2 x)) + 6 \cdot (\csc (2 x) \sec^{2} (6 x))$

Step 8.3.3

Reorder terms.

$- 2 \cot (2 x) \csc (2 x) \tan (6 x) + 6 \sec^{2} (6 x) \csc (2 x)$