Calculus Examples

$\tan (x) \cot (x)$

Step 1

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 $f (x) = \tan (x)$ and $g (x) = \cot (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Cancel the common factor of $\cos (x)$ .

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Step 4.2.4.1

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

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

Step 4.2.4.2

Cancel the common factor.

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

Step 4.2.4.3

Rewrite the expression.

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

Step 4.2.5

Multiply $\frac{1^{2}}{\cos (x)}$ by $\frac{1}{\sin (x)}$ .

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

Step 4.2.6

Separate fractions.

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

Step 4.2.7

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

$- \csc^{2} (x) \tan (x) + \frac{1^{2}}{\sin (x)} \sec (x)$

Step 4.2.8

Separate fractions.

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

Step 4.2.9

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

$- \csc^{2} (x) \tan (x) + \frac{1^{2}}{1} \csc (x) \sec (x)$

Step 4.2.10

Divide $1^{2}$ by $1$ .

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

Step 4.2.11

One to any power is one.

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

Step 4.2.12

Multiply $\csc (x)$ by $1$ .

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