Calculus Examples

$\cos (x y) = 9 + \sin (y)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (\cos (x y)) = \frac{d}{d y} (9 + \sin (y))$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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

Step 2.3.2

Multiply $x$ by $1$ .

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

Step 2.4

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

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

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $9 + \sin (y)$ with respect to $y$ is $\frac{d}{d y} [9] + \frac{d}{d y} [\sin (y)]$ .

$\frac{d}{d y} [9] + \frac{d}{d y} [\sin (y)]$

Step 3.1.2

Since $9$ is constant with respect to $y$ , the derivative of $9$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [\sin (y)]$

Step 3.2

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

$0 + \cos (y)$

Step 3.3

Add $0$ and $\cos (y)$ .

$\cos (y)$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

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.

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

Step 5.3.2.2

Cancel the common factor of $y$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Rewrite the expression.

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

Step 5.3.2.3

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

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Step 5.3.2.3.1

Cancel the common factor.

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

Step 5.3.2.3.2

Divide $x'$ by $1$ .

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

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

Separate fractions.

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

Step 5.3.3.1.2

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

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

Step 5.3.3.1.3

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

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

Step 5.3.3.1.4

Simplify.

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Step 5.3.3.1.4.1

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

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

Step 5.3.3.1.4.2

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

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

Step 5.3.3.1.5

Cancel the common factor of $1$ and $- 1$ .

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Step 5.3.3.1.5.1

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

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

Step 5.3.3.1.5.2

Move the negative in front of the fraction.

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

Step 5.3.3.1.6

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

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Step 5.3.3.1.6.1

Combine $\cos (y)$ and $\frac{1}{y}$ .

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

Step 5.3.3.1.6.2

Combine $\csc (x y)$ and $\frac{\cos (y)}{y}$ .

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

Step 5.3.3.1.7

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

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Step 5.3.3.1.7.1

Cancel the common factor.

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

Step 5.3.3.1.7.2

Rewrite the expression.

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

Step 5.3.3.1.8

Move the negative in front of the fraction.

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

Step 6

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

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