Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $g (x) = \cos (x y)$ .

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

Step 2.2

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 $g (x) = x y$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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 $g (x) = y$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $y$ by $1$ .

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

Step 2.7

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

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

Step 2.8

Multiply $\cos (x y)$ by $1$ .

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

Step 2.9

Simplify.

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Step 2.9.1

Apply the distributive property.

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

Step 2.9.2

Apply the distributive property.

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

Step 2.9.3

Combine terms.

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Step 2.9.3.1

Raise $x$ to the power of $1$ .

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

Step 2.9.3.2

Raise $x$ to the power of $1$ .

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

Step 2.9.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.9.3.4

Add $1$ and $1$ .

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

Step 2.9.4

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

$1 + \frac{d}{d x} [y]$

Step 3.3

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

$1 + y'$

Step 4

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

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

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 $- x^{2} \sin (x y) y' - x y \sin (x y) + \cos (x y)$ .

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

Step 5.2

Subtract $y'$ from both sides of the equation.

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

Step 5.3

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Factor $y'$ out of $- y'$ .

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

Step 5.4.3

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

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

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

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

Step 5.5.3.2

Combine the numerators over the common denominator.

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

Step 5.5.3.3

Combine the numerators over the common denominator.

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

Step 5.5.3.4

Factor $- 1$ out of $- x^{2} \sin (x y)$ .

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

Step 5.5.3.5

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

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

Step 5.5.3.6

Factor $- 1$ out of $- (x^{2} \sin (x y)) - 1 (1)$ .

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

Step 5.5.3.7

Rewrite negatives.

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Step 5.5.3.7.1

Rewrite $- (x^{2} \sin (x y) + 1)$ as $- 1 (x^{2} \sin (x y) + 1)$ .

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

Step 5.5.3.7.2

Move the negative in front of the fraction.

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

Step 6

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

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