Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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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 $f (x) = \cot (x)$ and $g (x) = x y$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

$- \csc^{2} (u) \frac{d}{d x} [x y] + \frac{d}{d x} [x y]$

Step 2.2.1.3

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

$- \csc^{2} (x y) \frac{d}{d x} [x y] + \frac{d}{d x} [x y]$

Step 2.2.2

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 $g (x) = y$ .

$- \csc^{2} (x y) (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [x y]$

Step 2.2.3

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

$- \csc^{2} (x y) (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [x y]$

Step 2.2.4

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

$- \csc^{2} (x y) (x y' + y \cdot 1) + \frac{d}{d x} [x y]$

Step 2.2.5

Multiply $y$ by $1$ .

$- \csc^{2} (x y) (x y' + y) + \frac{d}{d x} [x y]$

Step 2.3

Evaluate $\frac{d}{d x} [x y]$ .

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Step 2.3.1

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 $g (x) = y$ .

$- \csc^{2} (x y) (x y' + y) + x \frac{d}{d x} [y] + y \frac{d}{d x} [x]$

Step 2.3.2

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

$- \csc^{2} (x y) (x y' + y) + x y' + y \frac{d}{d x} [x]$

Step 2.3.3

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

$- \csc^{2} (x y) (x y' + y) + x y' + y \cdot 1$

Step 2.3.4

Multiply $y$ by $1$ .

$- \csc^{2} (x y) (x y' + y) + x y' + y$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$- \csc^{2} (x y) (x y') - \csc^{2} (x y) y + x y' + y$

Step 2.4.2

Reorder terms.

$- x \csc^{2} (x y) y' - y \csc^{2} (x y) + x y' + y$

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.

$- x \csc^{2} (x y) y' - y \csc^{2} (x y) + x y' + y = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Reorder factors in $- x \csc^{2} (x y) y' - y \csc^{2} (x y) + x y' + y$ .

$- x y' \csc^{2} (x y) - y \csc^{2} (x y) + x y' + y = 0$

Step 5.2

Simplify $- x y' \csc^{2} (x y) - y \csc^{2} (x y) + x y' + y$ .

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Step 5.2.1

Simplify the expression.

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Step 5.2.1.1

Move $x y'$ .

$- x y' \csc^{2} (x y) + x y' - y \csc^{2} (x y) + y = 0$

Step 5.2.1.2

Reorder $- x y' \csc^{2} (x y)$ and $x y'$ .

$x y' - x y' \csc^{2} (x y) - y \csc^{2} (x y) + y = 0$

Step 5.2.2

Factor $- x y'$ out of $x y'$ .

$- x y' \cdot - 1 - x y' \csc^{2} (x y) - y \csc^{2} (x y) + y = 0$

Step 5.2.3

Factor $- x y'$ out of $- x y' \csc^{2} (x y)$ .

$- x y' \cdot - 1 - x y' \csc^{2} (x y) - y \csc^{2} (x y) + y = 0$

Step 5.2.4

Factor $- x y'$ out of $- x y' \cdot - 1 - x y' \csc^{2} (x y)$ .

$- x y' (- 1 + \csc^{2} (x y)) - y \csc^{2} (x y) + y = 0$

Step 5.2.5

Rearrange terms.

$- x y' (\csc^{2} (x y) - 1) - y \csc^{2} (x y) + y = 0$

Step 5.2.6

Apply pythagorean identity.

$- x y' \cot^{2} (x y) - y \csc^{2} (x y) + y = 0$

Step 5.2.7

Reorder $- y \csc^{2} (x y)$ and $y$ .

$- x y' \cot^{2} (x y) + y - y \csc^{2} (x y) = 0$

Step 5.2.8

Rewrite $y$ as $1 y$ .

$- x y' \cot^{2} (x y) + 1 y - y \csc^{2} (x y) = 0$

Step 5.2.9

Factor $- y$ out of $1 y$ .

$- x y' \cot^{2} (x y) - y (- 1) - y \csc^{2} (x y) = 0$

Step 5.2.10

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

$- x y' \cot^{2} (x y) - y (- 1) - y (\csc^{2} (x y)) = 0$

Step 5.2.11

Factor $- y$ out of $- y (- 1) - y (\csc^{2} (x y))$ .

$- x y' \cot^{2} (x y) - y (- 1 + \csc^{2} (x y)) = 0$

Step 5.2.12

Rearrange terms.

$- x y' \cot^{2} (x y) - y (\csc^{2} (x y) - 1) = 0$

Step 5.2.13

Apply pythagorean identity.

$- x y' \cot^{2} (x y) - y \cot^{2} (x y) = 0$

Step 5.3

Add $y \cot^{2} (x y)$ to both sides of the equation.

$- x y' \cot^{2} (x y) = y \cot^{2} (x y)$

Step 5.4

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

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Step 5.4.1

Divide each term in $- x y' \cot^{2} (x y) = y \cot^{2} (x y)$ by $- x \cot^{2} (x y)$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Dividing two negative values results in a positive value.

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

Step 5.4.2.2

Cancel the common factor of $x$ .

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Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Rewrite the expression.

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

Step 5.4.2.3

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

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Step 5.4.2.3.1

Cancel the common factor.

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

Step 5.4.2.3.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

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

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Step 5.4.3.1.1

Cancel the common factor.

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

Step 5.4.3.1.2

Rewrite the expression.

$y' = \frac{y}{- x}$

Step 5.4.3.2

Move the negative in front of the fraction.

$y' = - \frac{y}{x}$

Step 6

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

$\frac{d y}{d x} = - \frac{y}{x}$