Calculus Examples

$\cot (x) \sec (x)$

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 4.2.1.1

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

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

Step 4.2.1.2

Cancel the common factors.

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

Step 4.2.2

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

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

Step 4.2.3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 4.2.3.1

Reorder $\csc (x)$ and $\tan (x)$ .

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

Step 4.2.3.2

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

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

Step 4.2.3.3

Cancel the common factors.

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

Step 4.2.4

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

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

Step 4.3

Factor $\sec (x)$ out of $\sec (x) - \csc^{2} (x) \sec (x)$ .

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Step 4.3.1

Multiply by $1$ .

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

Step 4.3.2

Factor $\sec (x)$ out of $- \csc^{2} (x) \sec (x)$ .

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

Step 4.3.3

Factor $\sec (x)$ out of $\sec (x) \cdot 1 + \sec (x) (- \csc^{2} (x))$ .

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

Step 4.4

Reorder $1$ and $- \csc^{2} (x)$ .

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

Step 4.5

Factor $- 1$ out of $- \csc^{2} (x)$ .

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

Step 4.6

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 4.7

Factor $- 1$ out of $- \csc^{2} (x) - 1 \cdot - 1$ .

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

Step 4.8

Apply pythagorean identity.

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

Step 4.9

Rewrite using the commutative property of multiplication.

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