Calculus Examples

$y = \frac{\sin (3 x)}{\tan (x)}$

Step 1

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

$\frac{d}{d x} [\frac{\sin (3 x)}{\frac{\sin (x)}{\cos (x)}}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$\frac{d}{d x} [\frac{\sin (3 x) \cos (x)}{\sin (x)}]$

Step 3

Convert from $\frac{\sin (3 x) \cos (x)}{\sin (x)}$ to $\sin (3 x) \cot (x)$ .

$\frac{d}{d x} [\sin (3 x) \cot (x)]$

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 $f (x) = \sin (3 x)$ and $g (x) = \cot (x)$ .

$\sin (3 x) \frac{d}{d x} [\cot (x)] + \cot (x) \frac{d}{d x} [\sin (3 x)]$

Step 5

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\sin (3 x) (- \csc^{2} (x)) + \cot (x) (\cos (u) \frac{d}{d x} [3 x])$

Step 6.3

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

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

Step 7

Differentiate.

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Step 7.1

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

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

Step 7.2

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

$\sin (3 x) (- \csc^{2} (x)) + \cot (x) \cos (3 x) (3 \cdot 1)$

Step 7.3

Simplify the expression.

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Step 7.3.1

Multiply $3$ by $1$ .

$\sin (3 x) (- \csc^{2} (x)) + \cot (x) \cos (3 x) \cdot 3$

Step 7.3.2

Move $3$ to the left of $\cot (x) \cos (3 x)$ .

$\sin (3 x) (- \csc^{2} (x)) + 3 \cdot (\cot (x) \cos (3 x))$

Step 8

Simplify.

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Step 8.1

Reorder terms.

$- \csc^{2} (x) \sin (3 x) + 3 \cos (3 x) \cot (x)$

Step 8.2

Simplify each term.

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Step 8.2.1

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

$- {(\frac{1}{\sin (x)})}^{2} \sin (3 x) + 3 \cos (3 x) \cot (x)$

Step 8.2.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

$- \frac{1^{2}}{\sin^{2} (x)} \sin (3 x) + 3 \cos (3 x) \cot (x)$

Step 8.2.3

One to any power is one.

$- \frac{1}{\sin^{2} (x)} \sin (3 x) + 3 \cos (3 x) \cot (x)$

Step 8.2.4

Combine $\sin (3 x)$ and $\frac{1}{\sin^{2} (x)}$ .

$- \frac{\sin (3 x)}{\sin^{2} (x)} + 3 \cos (3 x) \cot (x)$

Step 8.2.5

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

$- \frac{\sin (3 x)}{\sin^{2} (x)} + 3 \cos (3 x) \frac{\cos (x)}{\sin (x)}$

Step 8.2.6

Multiply $3 \cos (3 x) \frac{\cos (x)}{\sin (x)}$ .

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Step 8.2.6.1

Combine $\frac{\cos (x)}{\sin (x)}$ and $3$ .

$- \frac{\sin (3 x)}{\sin^{2} (x)} + \frac{\cos (x) \cdot 3}{\sin (x)} \cos (3 x)$

Step 8.2.6.2

Combine $\frac{\cos (x) \cdot 3}{\sin (x)}$ and $\cos (3 x)$ .

$- \frac{\sin (3 x)}{\sin^{2} (x)} + \frac{\cos (x) \cdot 3 \cos (3 x)}{\sin (x)}$

Step 8.2.7

Move $3$ to the left of $\cos (x)$ .

$- \frac{\sin (3 x)}{\sin^{2} (x)} + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3

Simplify each term.

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Step 8.3.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$- \frac{\sin (3 x)}{\sin (x) \sin (x)} + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.2

Separate fractions.

$- (\frac{1}{\sin (x)} \cdot \frac{\sin (3 x)}{\sin (x)}) + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.3

Rewrite $\frac{\sin (3 x)}{\sin (x)}$ as a product.

$- (\frac{1}{\sin (x)} (\sin (3 x) \frac{1}{\sin (x)})) + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.4

Write $\sin (3 x)$ as a fraction with denominator $1$ .

$- (\frac{1}{\sin (x)} (\frac{\sin (3 x)}{1} \cdot \frac{1}{\sin (x)})) + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.5

Simplify.

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Step 8.3.5.1

Divide $\sin (3 x)$ by $1$ .

$- (\frac{1}{\sin (x)} (\sin (3 x) \frac{1}{\sin (x)})) + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.5.2

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

$- (\frac{1}{\sin (x)} (\sin (3 x) \csc (x))) + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.6

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

$- (\csc (x) (\sin (3 x) \csc (x))) + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.7

Multiply $\csc (x) (\sin (3 x) \csc (x))$ .

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Step 8.3.7.1

Raise $\csc (x)$ to the power of $1$ .

$- (\csc^{1} (x) \csc (x) \sin (3 x)) + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.7.2

Raise $\csc (x)$ to the power of $1$ .

$- (\csc^{1} (x) \csc^{1} (x) \sin (3 x)) + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- ({\csc (x)}^{1 + 1} \sin (3 x)) + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.7.4

Add $1$ and $1$ .

$- (\csc^{2} (x) \sin (3 x)) + \frac{3 \cos (x) \cos (3 x)}{\sin (x)}$

Step 8.3.8

Separate fractions.

$- \csc^{2} (x) \sin (3 x) + \frac{3 \cos (3 x)}{1} \cdot \frac{\cos (x)}{\sin (x)}$

Step 8.3.9

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

$- \csc^{2} (x) \sin (3 x) + \frac{3 \cos (3 x)}{1} \cot (x)$

Step 8.3.10

Divide $3 \cos (3 x)$ by $1$ .

$- \csc^{2} (x) \sin (3 x) + 3 \cos (3 x) \cot (x)$