Calculus Examples

$\sin (x) = x (1 + \tan (y))$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$\cos (x)$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = x$ and $g (x) = 1 + \tan (y)$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

$x (0 + \frac{d}{d x} [\tan (y)]) + (1 + \tan (y)) \frac{d}{d x} [x]$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [\tan (y)]$ .

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

Step 3.3

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

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Step 3.3.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u$ with $y$ .

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

Step 3.4

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

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

Step 3.5

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

$x \sec^{2} (y) y' + (1 + \tan (y)) \cdot 1$

Step 3.6

Multiply $1 + \tan (y)$ by $1$ .

$x \sec^{2} (y) y' + 1 + \tan (y)$

Step 4

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

$\cos (x) = x \sec^{2} (y) y' + 1 + \tan (y)$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $x \sec^{2} (y) y' + 1 + \tan (y) = \cos (x)$ .

$x \sec^{2} (y) y' + 1 + \tan (y) = \cos (x)$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

$x y' \sec^{2} (y) + 1 + \tan (y) = \cos (x)$

Step 5.3

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.3.1

Subtract $1$ from both sides of the equation.

$x y' \sec^{2} (y) + \tan (y) = \cos (x) - 1$

Step 5.3.2

Subtract $\tan (y)$ from both sides of the equation.

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

Step 5.3.3

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

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

Step 5.3.4

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Rewrite the expression.

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

Step 5.4.2.2

Cancel the common factor of $\sec^{2} (y)$ .

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Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

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

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

Step 5.4.3.1.2

Separate fractions.

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

Step 5.4.3.1.3

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

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

Step 5.4.3.1.4

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

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

Step 5.4.3.1.5

Separate fractions.

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

Step 5.4.3.1.6

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

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

Step 5.4.3.1.7

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

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

Step 5.4.3.1.8

Multiply $\cos (y)$ by $1$ .

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

Step 5.4.3.1.9

Combine $\frac{1}{x}$ and $\cos (y)$ .

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

Step 5.4.3.1.10

Multiply $\frac{\cos (y)}{x} (\cos (x) \cos (y))$ .

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Step 5.4.3.1.10.1

Combine $\cos (x)$ and $\frac{\cos (y)}{x}$ .

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

Step 5.4.3.1.10.2

Combine $\frac{\cos (x) \cos (y)}{x}$ and $\cos (y)$ .

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

Step 5.4.3.1.10.3

Raise $\cos (y)$ to the power of $1$ .

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

Step 5.4.3.1.10.4

Raise $\cos (y)$ to the power of $1$ .

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

Step 5.4.3.1.10.5

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

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

Step 5.4.3.1.10.6

Add $1$ and $1$ .

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

Step 5.4.3.1.11

Simplify the denominator.

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Step 5.4.3.1.11.1

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

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

Step 5.4.3.1.11.2

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

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

Step 5.4.3.1.11.3

One to any power is one.

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

Step 5.4.3.1.12

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

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

Step 5.4.3.1.13

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{\cos (x) \cos^{2} (y)}{x} - \frac{\cos^{2} (y)}{x} + \frac{- \tan (y)}{x \sec^{2} (y)}$

Step 5.4.3.1.14

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

$y' = \frac{\cos (x) \cos^{2} (y)}{x} - \frac{\cos^{2} (y)}{x} + \frac{- \tan (y)}{x (\sec (y) \sec (y))}$

Step 5.4.3.1.15

Separate fractions.

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

Step 5.4.3.1.16

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

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

Step 5.4.3.1.17

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

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

Step 5.4.3.1.18

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

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

Step 5.4.3.1.19

Write $\cos (y)$ as a fraction with denominator $1$ .

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

Step 5.4.3.1.20

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

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Step 5.4.3.1.20.1

Cancel the common factor.

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

Step 5.4.3.1.20.2

Rewrite the expression.

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

Step 5.4.3.1.21

Separate fractions.

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

Step 5.4.3.1.22

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

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

Step 5.4.3.1.23

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

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

Step 5.4.3.1.24

Multiply $\cos (y)$ by $1$ .

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

Step 5.4.3.1.25

Move the negative in front of the fraction.

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

Step 5.4.3.1.26

Combine $\cos (y)$ and $\frac{1}{x}$ .

$y' = \frac{\cos (x) \cos^{2} (y)}{x} - \frac{\cos^{2} (y)}{x} - \frac{\cos (y)}{x} \sin (y)$

Step 5.4.3.1.27

Combine $\sin (y)$ and $\frac{\cos (y)}{x}$ .

$y' = \frac{\cos (x) \cos^{2} (y)}{x} - \frac{\cos^{2} (y)}{x} - \frac{\sin (y) \cos (y)}{x}$

Step 6

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

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