Calculus Examples

$2 \sin (x) + 3 \cos (y) - 3 \sin (x) \cos (y) + x = 7 π$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (2 \sin (x) + 3 \cos (y) - 3 \sin (x) \cos (y) + x) = \frac{d}{d x} (7 π)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $2 \sin (x) + 3 \cos (y) - 3 \sin (x) \cos (y) + x$ with respect to $x$ is $\frac{d}{d x} [2 \sin (x)] + \frac{d}{d x} [3 \cos (y)] + \frac{d}{d x} [- 3 \sin (x) \cos (y)] + \frac{d}{d x} [x]$ .

$\frac{d}{d x} [2 \sin (x)] + \frac{d}{d x} [3 \cos (y)] + \frac{d}{d x} [- 3 \sin (x) \cos (y)] + \frac{d}{d x} [x]$

Step 2.2

Evaluate $\frac{d}{d x} [2 \sin (x)]$ .

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Step 2.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \sin (x)$ with respect to $x$ is $2 \frac{d}{d x} [\sin (x)]$ .

$2 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [3 \cos (y)] + \frac{d}{d x} [- 3 \sin (x) \cos (y)] + \frac{d}{d x} [x]$

Step 2.2.2

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

$2 \cos (x) + \frac{d}{d x} [3 \cos (y)] + \frac{d}{d x} [- 3 \sin (x) \cos (y)] + \frac{d}{d x} [x]$

Step 2.3

Evaluate $\frac{d}{d x} [3 \cos (y)]$ .

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Step 2.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \cos (y)$ with respect to $x$ is $3 \frac{d}{d x} [\cos (y)]$ .

$2 \cos (x) + 3 \frac{d}{d x} [\cos (y)] + \frac{d}{d x} [- 3 \sin (x) \cos (y)] + \frac{d}{d x} [x]$

Step 2.3.2

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 $g (x) = y$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$2 \cos (x) + 3 (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [y]) + \frac{d}{d x} [- 3 \sin (x) \cos (y)] + \frac{d}{d x} [x]$

Step 2.3.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$2 \cos (x) + 3 (- \sin (u_{1}) \frac{d}{d x} [y]) + \frac{d}{d x} [- 3 \sin (x) \cos (y)] + \frac{d}{d x} [x]$

Step 2.3.2.3

Replace all occurrences of $u_{1}$ with $y$ .

$2 \cos (x) + 3 (- \sin (y) \frac{d}{d x} [y]) + \frac{d}{d x} [- 3 \sin (x) \cos (y)] + \frac{d}{d x} [x]$

Step 2.3.3

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

$2 \cos (x) + 3 (- \sin (y) y') + \frac{d}{d x} [- 3 \sin (x) \cos (y)] + \frac{d}{d x} [x]$

Step 2.3.4

Multiply $- 1$ by $3$ .

$2 \cos (x) - 3 \sin (y) y' + \frac{d}{d x} [- 3 \sin (x) \cos (y)] + \frac{d}{d x} [x]$

Step 2.4

Evaluate $\frac{d}{d x} [- 3 \sin (x) \cos (y)]$ .

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Step 2.4.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 \sin (x) \cos (y)$ with respect to $x$ is $- 3 \frac{d}{d x} [\sin (x) \cos (y)]$ .

$2 \cos (x) - 3 \sin (y) y' - 3 \frac{d}{d x} [\sin (x) \cos (y)] + \frac{d}{d x} [x]$

Step 2.4.2

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 $f (x) = \sin (x)$ and $g (x) = \cos (y)$ .

$2 \cos (x) - 3 \sin (y) y' - 3 (\sin (x) \frac{d}{d x} [\cos (y)] + \cos (y) \frac{d}{d x} [\sin (x)]) + \frac{d}{d x} [x]$

Step 2.4.3

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

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Step 2.4.3.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$2 \cos (x) - 3 \sin (y) y' - 3 (\sin (x) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [y]) + \cos (y) \frac{d}{d x} [\sin (x)]) + \frac{d}{d x} [x]$

Step 2.4.3.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$2 \cos (x) - 3 \sin (y) y' - 3 (\sin (x) (- \sin (u_{2}) \frac{d}{d x} [y]) + \cos (y) \frac{d}{d x} [\sin (x)]) + \frac{d}{d x} [x]$

Step 2.4.3.3

Replace all occurrences of $u_{2}$ with $y$ .

$2 \cos (x) - 3 \sin (y) y' - 3 (\sin (x) (- \sin (y) \frac{d}{d x} [y]) + \cos (y) \frac{d}{d x} [\sin (x)]) + \frac{d}{d x} [x]$

Step 2.4.4

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

$2 \cos (x) - 3 \sin (y) y' - 3 (\sin (x) (- \sin (y) y') + \cos (y) \frac{d}{d x} [\sin (x)]) + \frac{d}{d x} [x]$

Step 2.4.5

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

$2 \cos (x) - 3 \sin (y) y' - 3 (\sin (x) (- \sin (y) y') + \cos (y) \cos (x)) + \frac{d}{d x} [x]$

Step 2.5

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

$2 \cos (x) - 3 \sin (y) y' - 3 (\sin (x) (- \sin (y) y') + \cos (y) \cos (x)) + 1$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$2 \cos (x) - 3 \sin (y) y' - 3 (\sin (x) (- \sin (y) y')) - 3 (\cos (y) \cos (x)) + 1$

Step 2.6.2

Multiply $- 1$ by $- 3$ .

$2 \cos (x) - 3 \sin (y) y' + 3 \sin (x) \sin (y) y' - 3 \cos (y) \cos (x) + 1$

Step 2.6.3

Reorder terms.

$- 3 \sin (y) y' + 3 \sin (x) \sin (y) y' - 3 \cos (x) \cos (y) + 1 + 2 \cos (x)$

Step 3

Since $7 π$ is constant with respect to $x$ , the derivative of $7 π$ with respect to $x$ is $0$ .

$0$

Step 4

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

$- 3 \sin (y) y' + 3 \sin (x) \sin (y) y' - 3 \cos (x) \cos (y) + 1 + 2 \cos (x) = 0$

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 $- 3 \sin (y) y' + 3 \sin (x) \sin (y) y' - 3 \cos (x) \cos (y) + 1 + 2 \cos (x)$ .

$- 3 y' \sin (y) + 3 y' \sin (x) \sin (y) - 3 \cos (x) \cos (y) + 1 + 2 \cos (x) = 0$

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 $3 \cos (x) \cos (y)$ to both sides of the equation.

$- 3 y' \sin (y) + 3 y' \sin (x) \sin (y) + 1 + 2 \cos (x) = 3 \cos (x) \cos (y)$

Step 5.2.2

Subtract $1$ from both sides of the equation.

$- 3 y' \sin (y) + 3 y' \sin (x) \sin (y) + 2 \cos (x) = 3 \cos (x) \cos (y) - 1$

Step 5.2.3

Subtract $2 \cos (x)$ from both sides of the equation.

$- 3 y' \sin (y) + 3 y' \sin (x) \sin (y) = 3 \cos (x) \cos (y) - 1 - 2 \cos (x)$

Step 5.3

Factor $3 y' \sin (y)$ out of $- 3 y' \sin (y) + 3 y' \sin (x) \sin (y)$ .

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Step 5.3.1

Factor $3 y' \sin (y)$ out of $- 3 y' \sin (y)$ .

$3 y' \sin (y) \cdot - 1 + 3 y' \sin (x) \sin (y) = 3 \cos (x) \cos (y) - 1 - 2 \cos (x)$

Step 5.3.2

Factor $3 y' \sin (y)$ out of $3 y' \sin (x) \sin (y)$ .

$3 y' \sin (y) \cdot - 1 + 3 y' \sin (y) \sin (x) = 3 \cos (x) \cos (y) - 1 - 2 \cos (x)$

Step 5.3.3

Factor $3 y' \sin (y)$ out of $3 y' \sin (y) \cdot - 1 + 3 y' \sin (y) \sin (x)$ .

$3 y' \sin (y) (- 1 + \sin (x)) = 3 \cos (x) \cos (y) - 1 - 2 \cos (x)$

Step 5.4

Divide each term in $3 y' \sin (y) (- 1 + \sin (x)) = 3 \cos (x) \cos (y) - 1 - 2 \cos (x)$ by $3 \sin (y) (- 1 + \sin (x))$ and simplify.

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Step 5.4.1

Divide each term in $3 y' \sin (y) (- 1 + \sin (x)) = 3 \cos (x) \cos (y) - 1 - 2 \cos (x)$ by $3 \sin (y) (- 1 + \sin (x))$ .

$\frac{3 y' \sin (y) (- 1 + \sin (x))}{3 \sin (y) (- 1 + \sin (x))} = \frac{3 \cos (x) \cos (y)}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $3$ .

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Step 5.4.2.1.1

Cancel the common factor.

Step 5.4.2.1.2

Rewrite the expression.

$\frac{y' \sin (y) (- 1 + \sin (x))}{\sin (y) (- 1 + \sin (x))} = \frac{3 \cos (x) \cos (y)}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.2.2

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

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Step 5.4.2.2.1

Cancel the common factor.

Step 5.4.2.2.2

Rewrite the expression.

$\frac{y' (- 1 + \sin (x))}{- 1 + \sin (x)} = \frac{3 \cos (x) \cos (y)}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.2.3

Cancel the common factor of $- 1 + \sin (x)$ .

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Step 5.4.2.3.1

Cancel the common factor.

$\frac{y' (- 1 + \sin (x))}{- 1 + \sin (x)} = \frac{3 \cos (x) \cos (y)}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{3 \cos (x) \cos (y)}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

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

Cancel the common factor of $3$ .

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Step 5.4.3.1.1.1

Cancel the common factor.

$y' = \frac{3 \cos (x) \cos (y)}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.1.2

Rewrite the expression.

$y' = \frac{\cos (x) \cos (y)}{\sin (y) (- 1 + \sin (x))} + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.2

Separate fractions.

$y' = \frac{\cos (y)}{- 1 + \sin (x)} \cdot \frac{\cos (x)}{\sin (y)} + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.3

Rewrite $\frac{\cos (x)}{\sin (y)}$ as a product.

$y' = \frac{\cos (y)}{- 1 + \sin (x)} (\cos (x) \frac{1}{\sin (y)}) + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.4

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

$y' = \frac{\cos (y)}{- 1 + \sin (x)} (\frac{\cos (x)}{1} \cdot \frac{1}{\sin (y)}) + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.5

Simplify.

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Step 5.4.3.1.5.1

Divide $\cos (x)$ by $1$ .

$y' = \frac{\cos (y)}{- 1 + \sin (x)} (\cos (x) \frac{1}{\sin (y)}) + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.5.2

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

$y' = \frac{\cos (y)}{- 1 + \sin (x)} (\cos (x) \csc (y)) + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.6

Multiply $\frac{\cos (y)}{- 1 + \sin (x)} (\cos (x) \csc (y))$ .

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Step 5.4.3.1.6.1

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

$y' = \frac{\cos (x) \cos (y)}{- 1 + \sin (x)} \csc (y) + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.6.2

Combine $\frac{\cos (x) \cos (y)}{- 1 + \sin (x)}$ and $\csc (y)$ .

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} + \frac{- 1}{3 \sin (y) (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.7

Separate fractions.

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} + \frac{- 1}{3 (- 1 + \sin (x))} \cdot \frac{1}{\sin (y)} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.8

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

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} + \frac{- 1}{3 (- 1 + \sin (x))} \csc (y) + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.9

Move the negative in front of the fraction.

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{1}{3 (- 1 + \sin (x))} \csc (y) + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.10

Combine $\csc (y)$ and $\frac{1}{3 (- 1 + \sin (x))}$ .

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{\csc (y)}{3 (- 1 + \sin (x))} + \frac{- 2 \cos (x)}{3 \sin (y) (- 1 + \sin (x))}$

Step 5.4.3.1.11

Separate fractions.

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{\csc (y)}{3 (- 1 + \sin (x))} + \frac{- 2}{3 (- 1 + \sin (x))} \cdot \frac{\cos (x)}{\sin (y)}$

Step 5.4.3.1.12

Rewrite $\frac{\cos (x)}{\sin (y)}$ as a product.

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{\csc (y)}{3 (- 1 + \sin (x))} + \frac{- 2}{3 (- 1 + \sin (x))} (\cos (x) \frac{1}{\sin (y)})$

Step 5.4.3.1.13

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

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{\csc (y)}{3 (- 1 + \sin (x))} + \frac{- 2}{3 (- 1 + \sin (x))} (\frac{\cos (x)}{1} \cdot \frac{1}{\sin (y)})$

Step 5.4.3.1.14

Simplify.

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Step 5.4.3.1.14.1

Divide $\cos (x)$ by $1$ .

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{\csc (y)}{3 (- 1 + \sin (x))} + \frac{- 2}{3 (- 1 + \sin (x))} (\cos (x) \frac{1}{\sin (y)})$

Step 5.4.3.1.14.2

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

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{\csc (y)}{3 (- 1 + \sin (x))} + \frac{- 2}{3 (- 1 + \sin (x))} (\cos (x) \csc (y))$

Step 5.4.3.1.15

Move the negative in front of the fraction.

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{\csc (y)}{3 (- 1 + \sin (x))} - \frac{2}{3 (- 1 + \sin (x))} (\cos (x) \csc (y))$

Step 5.4.3.1.16

Multiply $- \frac{2}{3 (- 1 + \sin (x))} (\cos (x) \csc (y))$ .

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Step 5.4.3.1.16.1

Combine $\cos (x)$ and $\frac{2}{3 (- 1 + \sin (x))}$ .

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{\csc (y)}{3 (- 1 + \sin (x))} - \frac{\cos (x) \cdot 2}{3 (- 1 + \sin (x))} \csc (y)$

Step 5.4.3.1.16.2

Combine $\csc (y)$ and $\frac{\cos (x) \cdot 2}{3 (- 1 + \sin (x))}$ .

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{\csc (y)}{3 (- 1 + \sin (x))} - \frac{\csc (y) (\cos (x) \cdot 2)}{3 (- 1 + \sin (x))}$

Step 5.4.3.1.17

Move $2$ to the left of $\csc (y) \cos (x)$ .

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} - \frac{\csc (y)}{3 (- 1 + \sin (x))} - \frac{2 \csc (y) \cos (x)}{3 (- 1 + \sin (x))}$

Step 5.4.3.2

To write $\frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y' = \frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)} \cdot \frac{3}{3} - \frac{\csc (y)}{3 (- 1 + \sin (x))} - \frac{2 \csc (y) \cos (x)}{3 (- 1 + \sin (x))}$

Step 5.4.3.3

Write each expression with a common denominator of $(- 1 + \sin (x)) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.3.1

Multiply $\frac{\cos (x) \cos (y) \csc (y)}{- 1 + \sin (x)}$ by $\frac{3}{3}$ .

$y' = \frac{\cos (x) \cos (y) \csc (y) \cdot 3}{(- 1 + \sin (x)) \cdot 3} - \frac{\csc (y)}{3 (- 1 + \sin (x))} - \frac{2 \csc (y) \cos (x)}{3 (- 1 + \sin (x))}$

Step 5.4.3.3.2

Reorder the factors of $(- 1 + \sin (x)) \cdot 3$ .

$y' = \frac{\cos (x) \cos (y) \csc (y) \cdot 3}{3 (- 1 + \sin (x))} - \frac{\csc (y)}{3 (- 1 + \sin (x))} - \frac{2 \csc (y) \cos (x)}{3 (- 1 + \sin (x))}$

Step 5.4.3.4

Combine the numerators over the common denominator.

$y' = \frac{\cos (x) \cos (y) \csc (y) \cdot 3 - \csc (y)}{3 (- 1 + \sin (x))} - \frac{2 \csc (y) \cos (x)}{3 (- 1 + \sin (x))}$

Step 5.4.3.5

Simplify the numerator.

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Step 5.4.3.5.1

Factor $\csc (y)$ out of $\cos (x) \cos (y) \csc (y) \cdot 3 - \csc (y)$ .

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Step 5.4.3.5.1.1

Factor $\csc (y)$ out of $\cos (x) \cos (y) \csc (y) \cdot 3$ .

$y' = \frac{\csc (y) (\cos (x) \cos (y) \cdot 3) - \csc (y)}{3 (- 1 + \sin (x))} - \frac{2 \csc (y) \cos (x)}{3 (- 1 + \sin (x))}$

Step 5.4.3.5.1.2

Factor $\csc (y)$ out of $- \csc (y)$ .

$y' = \frac{\csc (y) (\cos (x) \cos (y) \cdot 3) + \csc (y) \cdot - 1}{3 (- 1 + \sin (x))} - \frac{2 \csc (y) \cos (x)}{3 (- 1 + \sin (x))}$

Step 5.4.3.5.1.3

Factor $\csc (y)$ out of $\csc (y) (\cos (x) \cos (y) \cdot 3) + \csc (y) \cdot - 1$ .

$y' = \frac{\csc (y) (\cos (x) \cos (y) \cdot 3 - 1)}{3 (- 1 + \sin (x))} - \frac{2 \csc (y) \cos (x)}{3 (- 1 + \sin (x))}$

Step 5.4.3.5.2

Move $3$ to the left of $\cos (x) \cos (y)$ .

$y' = \frac{\csc (y) (3 \cos (x) \cos (y) - 1)}{3 (- 1 + \sin (x))} - \frac{2 \csc (y) \cos (x)}{3 (- 1 + \sin (x))}$

Step 5.4.3.6

Simplify terms.

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Step 5.4.3.6.1

Combine the numerators over the common denominator.

$y' = \frac{\csc (y) (3 \cos (x) \cos (y) - 1) - 2 \csc (y) \cos (x)}{3 (- 1 + \sin (x))}$

Step 5.4.3.6.2

Factor $\csc (y)$ out of $\csc (y) (3 \cos (x) \cos (y) - 1) - 2 \csc (y) \cos (x)$ .

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Step 5.4.3.6.2.1

Factor $\csc (y)$ out of $- 2 \csc (y) \cos (x)$ .

$y' = \frac{\csc (y) (3 \cos (x) \cos (y) - 1) + \csc (y) (- 2 \cos (x))}{3 (- 1 + \sin (x))}$

Step 5.4.3.6.2.2

Factor $\csc (y)$ out of $\csc (y) (3 \cos (x) \cos (y) - 1) + \csc (y) (- 2 \cos (x))$ .

$y' = \frac{\csc (y) (3 \cos (x) \cos (y) - 1 - 2 \cos (x))}{3 (- 1 + \sin (x))}$

Step 5.4.3.6.3

Rewrite $- 1$ as $- 1 (1)$ .

$y' = \frac{\csc (y) (3 \cos (x) \cos (y) - 1 - 2 \cos (x))}{3 (- 1 (1) + \sin (x))}$

Step 5.4.3.6.4

Factor $- 1$ out of $\sin (x)$ .

$y' = \frac{\csc (y) (3 \cos (x) \cos (y) - 1 - 2 \cos (x))}{3 (- 1 (1) - 1 (- \sin (x)))}$

Step 5.4.3.6.5

Factor $- 1$ out of $- 1 (1) - 1 (- \sin (x))$ .

$y' = \frac{\csc (y) (3 \cos (x) \cos (y) - 1 - 2 \cos (x))}{3 (- 1 (1 - \sin (x)))}$

Step 5.4.3.6.6

Move the negative in front of the fraction.

$y' = - \frac{\csc (y) (3 \cos (x) \cos (y) - 1 - 2 \cos (x))}{3 (1 - \sin (x))}$

Step 6

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

$\frac{d y}{d x} = - \frac{\csc (y) (3 \cos (x) \cos (y) - 1 - 2 \cos (x))}{3 (1 - \sin (x))}$