Calculus Examples

$y = \frac{5}{\cos (x)} + \frac{1}{\tan (x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{5}{\cos (x)} + \frac{1}{\tan (x)})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $\frac{5}{\cos (x)} + \frac{1}{\tan (x)}$ with respect to $x$ is $\frac{d}{d x} [\frac{5}{\cos (x)}] + \frac{d}{d x} [\frac{1}{\tan (x)}]$ .

$\frac{d}{d x} [\frac{5}{\cos (x)}] + \frac{d}{d x} [\frac{1}{\tan (x)}]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{5}{\cos (x)}]$ .

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Step 3.2.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{\cos (x)}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{\cos (x)}]$ .

$5 \frac{d}{d x} [\frac{1}{\cos (x)}] + \frac{d}{d x} [\frac{1}{\tan (x)}]$

Step 3.2.2

Rewrite $\frac{1}{\cos (x)}$ as $\cos^{- 1} (x)$ .

$5 \frac{d}{d x} [\cos^{- 1} (x)] + \frac{d}{d x} [\frac{1}{\tan (x)}]$

Step 3.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = \cos (x)$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

$5 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [\frac{1}{\tan (x)}]$

Step 3.2.3.2

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

$5 (- u^{- 2} \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [\frac{1}{\tan (x)}]$

Step 3.2.3.3

Replace all occurrences of $u$ with $\cos (x)$ .

$5 (- \cos^{- 2} (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [\frac{1}{\tan (x)}]$

Step 3.2.4

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

$5 (- \cos^{- 2} (x) (- \sin (x))) + \frac{d}{d x} [\frac{1}{\tan (x)}]$

Step 3.2.5

Multiply $- 1$ by $- 1$ .

$5 (1 \cos^{- 2} (x) \sin (x)) + \frac{d}{d x} [\frac{1}{\tan (x)}]$

Step 3.2.6

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

$5 \cos^{- 2} (x) \sin (x) + \frac{d}{d x} [\frac{1}{\tan (x)}]$

Step 3.3

Evaluate $\frac{d}{d x} [\frac{1}{\tan (x)}]$ .

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Step 3.3.1

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

$5 \cos^{- 2} (x) \sin (x) + \frac{d}{d x} [\frac{1}{\frac{\sin (x)}{\cos (x)}}]$

Step 3.3.2

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

$5 \cos^{- 2} (x) \sin (x) + \frac{d}{d x} [\frac{\cos (x)}{\sin (x)}]$

Step 3.3.3

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

$5 \cos^{- 2} (x) \sin (x) + \frac{d}{d x} [\cot (x)]$

Step 3.3.4

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

$5 \cos^{- 2} (x) \sin (x) - \csc^{2} (x)$

Step 3.4

Simplify.

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Step 3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 \frac{1}{\cos^{2} (x)} \sin (x) - \csc^{2} (x)$

Step 3.4.2

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

$5 \sec^{2} (x) \sin (x) - \csc^{2} (x)$

Step 3.4.3

Simplify each term.

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Step 3.4.3.1

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

$5 {(\frac{1}{\cos (x)})}^{2} \sin (x) - \csc^{2} (x)$

Step 3.4.3.2

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

$5 \frac{1^{2}}{\cos^{2} (x)} \sin (x) - \csc^{2} (x)$

Step 3.4.3.3

One to any power is one.

$5 \frac{1}{\cos^{2} (x)} \sin (x) - \csc^{2} (x)$

Step 3.4.3.4

Combine $5$ and $\frac{1}{\cos^{2} (x)}$ .

$\frac{5}{\cos^{2} (x)} \sin (x) - \csc^{2} (x)$

Step 3.4.3.5

Combine $\frac{5}{\cos^{2} (x)}$ and $\sin (x)$ .

$\frac{5 \sin (x)}{\cos^{2} (x)} - \csc^{2} (x)$

Step 3.4.3.6

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

$\frac{5 \sin (x)}{\cos^{2} (x)} - {(\frac{1}{\sin (x)})}^{2}$

Step 3.4.3.7

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

$\frac{5 \sin (x)}{\cos^{2} (x)} - \frac{1^{2}}{\sin^{2} (x)}$

Step 3.4.3.8

One to any power is one.

$\frac{5 \sin (x)}{\cos^{2} (x)} - \frac{1}{\sin^{2} (x)}$

Step 3.4.4

Simplify each term.

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Step 3.4.4.1

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

$\frac{5 \sin (x)}{\cos (x) \cos (x)} - \frac{1}{\sin^{2} (x)}$

Step 3.4.4.2

Separate fractions.

$\frac{5}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} - \frac{1}{\sin^{2} (x)}$

Step 3.4.4.3

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

$\frac{5}{\cos (x)} \tan (x) - \frac{1}{\sin^{2} (x)}$

Step 3.4.4.4

Separate fractions.

$\frac{5}{1} \cdot \frac{1}{\cos (x)} \tan (x) - \frac{1}{\sin^{2} (x)}$

Step 3.4.4.5

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

$\frac{5}{1} \sec (x) \tan (x) - \frac{1}{\sin^{2} (x)}$

Step 3.4.4.6

Divide $5$ by $1$ .

$5 \sec (x) \tan (x) - \frac{1}{\sin^{2} (x)}$

Step 3.4.4.7

Rewrite $1$ as $1^{2}$ .

$5 \sec (x) \tan (x) - \frac{1^{2}}{\sin^{2} (x)}$

Step 3.4.4.8

Rewrite $\frac{1^{2}}{\sin^{2} (x)}$ as ${(\frac{1}{\sin (x)})}^{2}$ .

$5 \sec (x) \tan (x) - {(\frac{1}{\sin (x)})}^{2}$

Step 3.4.4.9

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

$5 \sec (x) \tan (x) - \csc^{2} (x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 5 \sec (x) \tan (x) - \csc^{2} (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 5 \sec (x) \tan (x) - \csc^{2} (x)$