Calculus Examples

$x y = \cot (x y)$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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 $g (x) = y$ .

$x \frac{d}{d x} [y] + y \frac{d}{d x} [x]$

Step 2.2

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

$x y' + y \frac{d}{d x} [x]$

Step 2.3

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

$x y' + y \cdot 1$

Step 2.4

Multiply $y$ by $1$ .

$x y' + y$

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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 $g (x) = y$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply $y$ by $1$ .

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

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

Reorder terms.

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the right side.

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Step 5.1.1

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

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

Step 5.2

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

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

Step 5.3

Subtract $y$ from both sides of the equation.

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $x$ .

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Rewrite the expression.

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

Step 5.5.2.2

Cancel the common factor of $1 + \csc^{2} (x y)$ .

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Step 5.5.2.2.1

Cancel the common factor.

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

Step 5.5.2.2.2

Divide $y'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Move the negative in front of the fraction.

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

Step 5.5.3.1.2

Move the negative in front of the fraction.

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

Step 6

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

$\frac{d y}{d x} = - \frac{y \csc^{2} (x y)}{x (1 + \csc^{2} (x y))} - \frac{y}{x (1 + \csc^{2} (x y))}$