Calculus Examples

$f (x) = \frac{\csc (3 x)}{\cot (3 x)}$

Step 1

Rewrite $\cot (3 x)$ in terms of sines and cosines.

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

Step 2

Rewrite $\csc (3 x)$ in terms of sines and cosines.

$\frac{d}{d x} [\frac{\frac{1}{\sin (3 x)}}{\frac{\cos (3 x)}{\sin (3 x)}}]$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (3 x)}{\sin (3 x)}$ .

$\frac{d}{d x} [\frac{1}{\sin (3 x)} \cdot \frac{\sin (3 x)}{\cos (3 x)}]$

Step 4

Cancel the common factor of $\sin (3 x)$ .

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Step 4.1

Cancel the common factor.

$\frac{d}{d x} [\frac{1}{\sin (3 x)} \cdot \frac{\sin (3 x)}{\cos (3 x)}]$

Step 4.2

Rewrite the expression.

$\frac{d}{d x} [\frac{1}{\cos (3 x)}]$

Step 5

Convert from $\frac{1}{\cos (3 x)}$ to $\sec (3 x)$ .

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

Step 6

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

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Step 6.1

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

$\frac{d}{d u} [\sec (u)] \frac{d}{d x} [3 x]$

Step 6.2

The derivative of $\sec (u)$ with respect to $u$ is $\sec (u) \tan (u)$ .

$\sec (u) \tan (u) \frac{d}{d x} [3 x]$

Step 6.3

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

$\sec (3 x) \tan (3 x) \frac{d}{d x} [3 x]$

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 $3 \frac{d}{d x} [x]$ .

$\sec (3 x) \tan (3 x) (3 \frac{d}{d x} [x])$

Step 7.2

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

$\sec (3 x) \tan (3 x) (3 \cdot 1)$

Step 7.3

Simplify the expression.

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Step 7.3.1

Multiply $3$ by $1$ .

$\sec (3 x) \tan (3 x) \cdot 3$

Step 7.3.2

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

$3 \sec (3 x) \tan (3 x)$