Calculus Examples

$\sin (x) - \cos (y) - 2 = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\sin (x) - \cos (y) - 2) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (y)$ with respect to $x$ is $- \frac{d}{d x} [\cos (y)]$ .

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

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 $u$ as $y$ .

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $- 1$ by $- 1$ .

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

Step 2.3.5

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

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

Step 2.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\cos (x) + \sin (y) y' + 0$

Step 2.5

Simplify.

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Step 2.5.1

Add $\cos (x) + \sin (y) y'$ and $0$ .

$\cos (x) + \sin (y) y'$

Step 2.5.2

Reorder terms.

$\sin (y) y' + \cos (x)$

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.

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

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 $\sin (y) y' + \cos (x)$ .

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

Step 5.2

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

$y' \sin (y) = - \cos (x)$

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Separate fractions.

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

Step 5.3.3.2

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

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

Step 5.3.3.3

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

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

Step 5.3.3.4

Simplify.

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Step 5.3.3.4.1

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

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

Step 5.3.3.4.2

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

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

Step 5.3.3.5

Divide $- 1$ by $1$ .

$y' = - \cos (x) \csc (y)$

Step 6

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

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