Calculus Examples

$\cos (x y) = 2 x$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 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 $g (x) = y$ .

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

Step 2.3

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

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

Step 2.4

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

$- \sin (x y) (x y' + y \cdot 1)$

Step 2.5

Multiply $y$ by $1$ .

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$2 \frac{d}{d x} [x]$

Step 3.2

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

$2 \cdot 1$

Step 3.3

Multiply $2$ by $1$ .

$2$

Step 4

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

$- x \sin (x y) y' - y \sin (x y) = 2$

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

$- x y' \sin (x y) - y \sin (x y) = 2$

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

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.

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

Step 5.3.2.2

Cancel the common factor of $x$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Rewrite the expression.

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

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.

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

Step 5.3.2.3.2

Divide $y'$ by $1$ .

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

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.

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

Step 5.3.3.1.2

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

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

Step 5.3.3.1.3

Move the negative in front of the fraction.

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

Step 5.3.3.1.4

Combine $\csc (x y)$ and $\frac{2}{x}$ .

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

Step 5.3.3.1.5

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

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

Step 5.3.3.1.6

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

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Step 5.3.3.1.6.1

Cancel the common factor.

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

Step 5.3.3.1.6.2

Rewrite the expression.

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

Step 5.3.3.1.7

Move the negative in front of the fraction.

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

Step 6

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

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