Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = \frac{1}{y}$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $\frac{1}{y}$ .

$\frac{d}{d u} [\sec (u)] \frac{d}{d x} [\frac{1}{y}]$

Step 3.1.2

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

$\sec (u) \tan (u) \frac{d}{d x} [\frac{1}{y}]$

Step 3.1.3

Replace all occurrences of $u$ with $\frac{1}{y}$ .

$\sec (\frac{1}{y}) \tan (\frac{1}{y}) \frac{d}{d x} [\frac{1}{y}]$

Step 3.2

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 $f (x) = 1$ and $g (x) = y$ .

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

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Multiply $- 1$ by $1$ .

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

Step 3.3.2

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

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

Step 3.3.3

Simplify the expression.

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Step 3.3.3.1

Multiply $y$ by $0$ .

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

Step 3.3.3.2

Subtract $\frac{d}{d x} [y]$ from $0$ .

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

Step 3.3.3.3

Move the negative in front of the fraction.

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

Step 3.4

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

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

Step 3.5

Combine $\sec (\frac{1}{y})$ and $\frac{y'}{y^{2}}$ .

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

Step 3.6

Combine $\tan (\frac{1}{y})$ and $\frac{\sec (\frac{1}{y}) y'}{y^{2}}$ .

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

Step 4

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

$1 = - \frac{\tan (\frac{1}{y}) \sec (\frac{1}{y}) y'}{y^{2}}$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $- \frac{\tan (\frac{1}{y}) \sec (\frac{1}{y}) y'}{y^{2}} = 1$ .

$- \frac{\tan (\frac{1}{y}) \sec (\frac{1}{y}) y'}{y^{2}} = 1$

Step 5.2

Divide each term in $- \frac{\tan (\frac{1}{y}) \sec (\frac{1}{y}) y'}{y^{2}} = 1$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- \frac{\tan (\frac{1}{y}) \sec (\frac{1}{y}) y'}{y^{2}} = 1$ by $- 1$ .

$\frac{- \frac{\tan (\frac{1}{y}) \sec (\frac{1}{y}) y'}{y^{2}}}{- 1} = \frac{1}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{\tan (\frac{1}{y}) \sec (\frac{1}{y}) y'}{y^{2}}}{1} = \frac{1}{- 1}$

Step 5.2.2.2

Simplify the expression.

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Step 5.2.2.2.1

Divide $\frac{\tan (\frac{1}{y}) \sec (\frac{1}{y}) y'}{y^{2}}$ by $1$ .

$\frac{\tan (\frac{1}{y}) \sec (\frac{1}{y}) y'}{y^{2}} = \frac{1}{- 1}$

Step 5.2.2.2.2

Reorder factors in $\frac{\tan (\frac{1}{y}) \sec (\frac{1}{y}) y'}{y^{2}}$ .

$\frac{y' \tan (\frac{1}{y}) \sec (\frac{1}{y})}{y^{2}} = \frac{1}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $1$ by $- 1$ .

$\frac{y' \tan (\frac{1}{y}) \sec (\frac{1}{y})}{y^{2}} = - 1$

Step 5.3

Multiply both sides by $y^{2}$ .

$\frac{y' \tan (\frac{1}{y}) \sec (\frac{1}{y})}{y^{2}} y^{2} = - y^{2}$

Step 5.4

Simplify the left side.

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Step 5.4.1

Cancel the common factor of $y^{2}$ .

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Step 5.4.1.1

Cancel the common factor.

$\frac{y' \tan (\frac{1}{y}) \sec (\frac{1}{y})}{y^{2}} y^{2} = - y^{2}$

Step 5.4.1.2

Rewrite the expression.

$y' \tan (\frac{1}{y}) \sec (\frac{1}{y}) = - y^{2}$

Step 5.5

Divide each term in $y' \tan (\frac{1}{y}) \sec (\frac{1}{y}) = - y^{2}$ by $\tan (\frac{1}{y}) \sec (\frac{1}{y})$ and simplify.

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Step 5.5.1

Divide each term in $y' \tan (\frac{1}{y}) \sec (\frac{1}{y}) = - y^{2}$ by $\tan (\frac{1}{y}) \sec (\frac{1}{y})$ .

$\frac{y' \tan (\frac{1}{y}) \sec (\frac{1}{y})}{\tan (\frac{1}{y}) \sec (\frac{1}{y})} = \frac{- y^{2}}{\tan (\frac{1}{y}) \sec (\frac{1}{y})}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $\tan (\frac{1}{y})$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{y' \tan (\frac{1}{y}) \sec (\frac{1}{y})}{\tan (\frac{1}{y}) \sec (\frac{1}{y})} = \frac{- y^{2}}{\tan (\frac{1}{y}) \sec (\frac{1}{y})}$

Step 5.5.2.1.2

Rewrite the expression.

$\frac{y' \sec (\frac{1}{y})}{\sec (\frac{1}{y})} = \frac{- y^{2}}{\tan (\frac{1}{y}) \sec (\frac{1}{y})}$

Step 5.5.2.2

Cancel the common factor of $\sec (\frac{1}{y})$ .

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Step 5.5.2.2.1

Cancel the common factor.

$\frac{y' \sec (\frac{1}{y})}{\sec (\frac{1}{y})} = \frac{- y^{2}}{\tan (\frac{1}{y}) \sec (\frac{1}{y})}$

Step 5.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- y^{2}}{\tan (\frac{1}{y}) \sec (\frac{1}{y})}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

$y' = - \frac{y^{2}}{\tan (\frac{1}{y}) \sec (\frac{1}{y})}$

Step 5.5.3.2

Separate fractions.

$y' = - (\frac{y^{2}}{\sec (\frac{1}{y})} \cdot \frac{1}{\tan (\frac{1}{y})})$

Step 5.5.3.3

Rewrite $\tan (\frac{1}{y})$ in terms of sines and cosines.

$y' = - (\frac{y^{2}}{\sec (\frac{1}{y})} \cdot \frac{1}{\frac{\sin (\frac{1}{y})}{\cos (\frac{1}{y})}})$

Step 5.5.3.4

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (\frac{1}{y})}{\cos (\frac{1}{y})}$ .

$y' = - (\frac{y^{2}}{\sec (\frac{1}{y})} \cdot \frac{\cos (\frac{1}{y})}{\sin (\frac{1}{y})})$

Step 5.5.3.5

Convert from $\frac{\cos (\frac{1}{y})}{\sin (\frac{1}{y})}$ to $\cot (\frac{1}{y})$ .

$y' = - (\frac{y^{2}}{\sec (\frac{1}{y})} \cot (\frac{1}{y}))$

Step 5.5.3.6

Separate fractions.

$y' = - (\frac{y^{2}}{1} \cdot \frac{1}{\sec (\frac{1}{y})} \cot (\frac{1}{y}))$

Step 5.5.3.7

Rewrite $\sec (\frac{1}{y})$ in terms of sines and cosines.

$y' = - (\frac{y^{2}}{1} \cdot \frac{1}{\frac{1}{\cos (\frac{1}{y})}} \cot (\frac{1}{y}))$

Step 5.5.3.8

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (\frac{1}{y})}$ .

$y' = - (\frac{y^{2}}{1} (1 \cos (\frac{1}{y})) \cot (\frac{1}{y}))$

Step 5.5.3.9

Multiply $\cos (\frac{1}{y})$ by $1$ .

$y' = - (\frac{y^{2}}{1} \cos (\frac{1}{y}) \cot (\frac{1}{y}))$

Step 5.5.3.10

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

$y' = - y^{2} \cos (\frac{1}{y}) \cot (\frac{1}{y})$

Step 6

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

$\frac{d y}{d x} = - y^{2} \cos (\frac{1}{y}) \cot (\frac{1}{y})$