Calculus Examples

$x + \cos (x + y) = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x + \cos (x + y)) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

$1 - \sin (x + y) (1 + y')$

Step 2.3

Reorder terms.

$- \sin (x + y) (1 + y') + 1$

Step 3

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

$0$

Step 4

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

$- \sin (x + y) (1 + y') + 1 = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $- \sin (x + y) (1 + y') + 1$ .

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Step 5.1.1.1

Simplify each term.

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Step 5.1.1.1.1

Apply the distributive property.

$- \sin (x + y) \cdot 1 - \sin (x + y) y' + 1 = 0$

Step 5.1.1.1.2

Multiply $- 1$ by $1$ .

$- \sin (x + y) - \sin (x + y) y' + 1 = 0$

Step 5.1.1.2

Reorder factors in $- \sin (x + y) - \sin (x + y) y' + 1$ .

$- \sin (x + y) - y' \sin (x + y) + 1 = 0$

Step 5.2

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

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Step 5.2.1

Add $\sin (x + y)$ to both sides of the equation.

$- y' \sin (x + y) + 1 = \sin (x + y)$

Step 5.2.2

Subtract $1$ from both sides of the equation.

$- y' \sin (x + y) = \sin (x + y) - 1$

Step 5.3

Divide each term in $- y' \sin (x + y) = \sin (x + y) - 1$ by $- \sin (x + y)$ and simplify.

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Step 5.3.1

Divide each term in $- y' \sin (x + y) = \sin (x + y) - 1$ by $- \sin (x + y)$ .

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Dividing two negative values results in a positive value.

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

Step 5.3.2.2

Cancel the common factor of $\sin (x + y)$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $\sin (x + y)$ .

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Step 5.3.3.1.1.1

Cancel the common factor.

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

Step 5.3.3.1.1.2

Rewrite the expression.

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

Step 5.3.3.1.1.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

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

Step 5.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 5.3.3.1.3

Dividing two negative values results in a positive value.

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

Step 5.3.3.1.4

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

$y' = - 1 + \csc (x + y)$

Step 6

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

$\frac{d y}{d x} = - 1 + \csc (x + y)$