Calculus Examples

$y = \frac{1 - \sec (x)}{\tan (x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1 - \sec (x)}{\tan (x)})$

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

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1 - \sec (x)$ and $g (x) = \tan (x)$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{\tan (x) (0 + \frac{d}{d x} [- \sec (x)]) - (1 - \sec (x)) \frac{d}{d x} [\tan (x)]}{\tan^{2} (x)}$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [- \sec (x)]$ .

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

Step 3.2.4

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Add $1$ and $1$ .

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

Step 3.8

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

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

Step 3.9

Simplify.

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Step 3.9.1

Apply the distributive property.

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

Step 3.9.2

Apply the distributive property.

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

Step 3.9.3

Simplify the numerator.

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Step 3.9.3.1

Simplify each term.

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Step 3.9.3.1.1

Multiply $- 1$ by $1$ .

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

Step 3.9.3.1.2

Rewrite $- 1 \sec^{2} (x)$ as $- \sec^{2} (x)$ .

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

Step 3.9.3.1.3

Multiply $\sec (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Step 3.9.3.1.3.1

Move $\sec^{2} (x)$ .

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

Step 3.9.3.1.3.2

Multiply $\sec^{2} (x)$ by $\sec (x)$ .

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Step 3.9.3.1.3.2.1

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

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

Step 3.9.3.1.3.2.2

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

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

Step 3.9.3.1.3.3

Add $2$ and $1$ .

$\frac{- \sec (x) \tan^{2} (x) - \sec^{2} (x) - - \sec^{3} (x)}{\tan^{2} (x)}$

Step 3.9.3.1.4

Multiply $- - \sec^{3} (x)$ .

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Step 3.9.3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- \sec (x) \tan^{2} (x) - \sec^{2} (x) + 1 \sec^{3} (x)}{\tan^{2} (x)}$

Step 3.9.3.1.4.2

Multiply $\sec^{3} (x)$ by $1$ .

$\frac{- \sec (x) \tan^{2} (x) - \sec^{2} (x) + \sec^{3} (x)}{\tan^{2} (x)}$

Step 3.9.3.2

Factor $\sec (x)$ out of $- \sec (x) \tan^{2} (x) - \sec^{2} (x) + \sec^{3} (x)$ .

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Step 3.9.3.2.1

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

$\frac{\sec (x) (- \tan^{2} (x)) - \sec^{2} (x) + \sec^{3} (x)}{\tan^{2} (x)}$

Step 3.9.3.2.2

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

$\frac{\sec (x) (- \tan^{2} (x)) + \sec (x) (- \sec (x)) + \sec^{3} (x)}{\tan^{2} (x)}$

Step 3.9.3.2.3

Factor $\sec (x)$ out of $\sec^{3} (x)$ .

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

Step 3.9.3.2.4

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

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

Step 3.9.3.2.5

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

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

Step 3.9.3.3

Move $\sec^{2} (x)$ .

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

Step 3.9.3.4

Reorder $- \tan^{2} (x)$ and $\sec^{2} (x)$ .

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

Step 3.9.3.5

Apply pythagorean identity.

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

Step 3.9.3.6

Apply the distributive property.

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

Step 3.9.3.7

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

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

Step 3.9.3.8

Rewrite using the commutative property of multiplication.

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

Step 3.9.3.9

Multiply $- \sec (x) \sec (x)$ .

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Step 3.9.3.9.1

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

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

Step 3.9.3.9.2

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

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

Step 3.9.3.9.3

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

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

Step 3.9.3.9.4

Add $1$ and $1$ .

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

Step 3.9.4

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

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Step 3.9.4.1

Multiply by $1$ .

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

Step 3.9.4.2

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

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

Step 3.9.4.3

Factor $\sec (x)$ out of $\sec (x) \cdot 1 + \sec (x) (- \sec (x))$ .

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

Step 3.9.5

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

$\frac{\sec (x) (1 - \sec (x))}{\tan (x) \tan (x)}$

Step 3.9.6

Separate fractions.

$\frac{1 - \sec (x)}{\tan (x)} \cdot \frac{\sec (x)}{\tan (x)}$

Step 3.9.7

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

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

Step 3.9.8

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

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

Step 3.9.9

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

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

Step 3.9.10

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

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Step 3.9.10.1

Cancel the common factor.

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

Step 3.9.10.2

Rewrite the expression.

$\frac{1 - \sec (x)}{\tan (x)} \cdot \frac{1}{\sin (x)}$

Step 3.9.11

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

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

Step 3.9.12

Combine $\frac{1 - \sec (x)}{\tan (x)}$ and $\csc (x)$ .

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

Step 3.9.13

Separate fractions.

$\frac{1 - \sec (x)}{1} \cdot \frac{\csc (x)}{\tan (x)}$

Step 3.9.14

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

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

Step 3.9.15

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

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

Step 3.9.16

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

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

Step 3.9.17

Simplify.

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Step 3.9.17.1

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

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

Step 3.9.17.2

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

$\frac{1 - \sec (x)}{1} (\csc (x) \cot (x))$

Step 3.9.18

Divide $1 - \sec (x)$ by $1$ .

$(1 - \sec (x)) (\csc (x) \cot (x))$

Step 3.9.19

Apply the distributive property.

$1 (\csc (x) \cot (x)) - \sec (x) (\csc (x) \cot (x))$

Step 3.9.20

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

$\csc (x) \cot (x) - \sec (x) \csc (x) \cot (x)$

Step 4

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

$y' = \csc (x) \cot (x) - \sec (x) \csc (x) \cot (x)$

Step 5

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

$\frac{d y}{d x} = \csc (x) \cot (x) - \sec (x) \csc (x) \cot (x)$