Calculus Examples

$6 x e^{x y} = 5$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (6 x e^{x y}) = \frac{d}{d x} (5)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x e^{x y}$ with respect to $x$ is $6 \frac{d}{d x} [x e^{x y}]$ .

$6 \frac{d}{d x} [x e^{x y}]$

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 $g (x) = e^{x y}$ .

$6 (x \frac{d}{d x} [e^{x y}] + e^{x y} \frac{d}{d x} [x])$

Step 2.3

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

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Step 2.3.1

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

$6 (x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x y]) + e^{x y} \frac{d}{d x} [x])$

Step 2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$6 (x (e^{u} \frac{d}{d x} [x y]) + e^{x y} \frac{d}{d x} [x])$

Step 2.3.3

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

$6 (x (e^{x y} \frac{d}{d x} [x y]) + e^{x y} \frac{d}{d x} [x])$

Step 2.4

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 $g (x) = y$ .

$6 (x (e^{x y} (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])) + e^{x y} \frac{d}{d x} [x])$

Step 2.5

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

$6 (x (e^{x y} (x y' + y \frac{d}{d x} [x])) + e^{x y} \frac{d}{d x} [x])$

Step 2.6

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

$6 (x (e^{x y} (x y' + y \cdot 1)) + e^{x y} \frac{d}{d x} [x])$

Step 2.7

Multiply $y$ by $1$ .

$6 (x (e^{x y} (x y' + y)) + e^{x y} \frac{d}{d x} [x])$

Step 2.8

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

$6 (x (e^{x y} (x y' + y)) + e^{x y} \cdot 1)$

Step 2.9

Multiply $e^{x y}$ by $1$ .

$6 (x (e^{x y} (x y' + y)) + e^{x y})$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$6 (x (e^{x y} (x y') + e^{x y} y) + e^{x y})$

Step 2.10.2

Apply the distributive property.

$6 (x (e^{x y} (x y')) + x (e^{x y} y) + e^{x y})$

Step 2.10.3

Apply the distributive property.

$6 (x (e^{x y} (x y'))) + 6 (x (e^{x y} y)) + 6 e^{x y}$

Step 2.10.4

Combine terms.

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Step 2.10.4.1

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

$6 (x^{1} x (e^{x y} (y'))) + 6 (x (e^{x y} y)) + 6 e^{x y}$

Step 2.10.4.2

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

$6 (x^{1} x^{1} (e^{x y} (y'))) + 6 (x (e^{x y} y)) + 6 e^{x y}$

Step 2.10.4.3

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

$6 (x^{1 + 1} (e^{x y} (y'))) + 6 (x (e^{x y} y)) + 6 e^{x y}$

Step 2.10.4.4

Add $1$ and $1$ .

$6 x^{2} (e^{x y} (y')) + 6 x (e^{x y} y) + 6 e^{x y}$

Step 2.10.5

Reorder terms.

$6 e^{x y} x^{2} y' + 6 e^{x y} x y + 6 e^{x y}$

Step 3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0$

Step 4

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

$6 e^{x y} x^{2} y' + 6 e^{x y} x y + 6 e^{x y} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Reorder factors in $6 e^{x y} x^{2} y' + 6 e^{x y} x y + 6 e^{x y}$ .

$6 x^{2} y' e^{x y} + 6 x y e^{x y} + 6 e^{x y} = 0$

Step 5.2

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

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Step 5.2.1

Subtract $6 x y e^{x y}$ from both sides of the equation.

$6 x^{2} y' e^{x y} + 6 e^{x y} = - 6 x y e^{x y}$

Step 5.2.2

Subtract $6 e^{x y}$ from both sides of the equation.

$6 x^{2} y' e^{x y} = - 6 x y e^{x y} - 6 e^{x y}$

Step 5.3

Divide each term in $6 x^{2} y' e^{x y} = - 6 x y e^{x y} - 6 e^{x y}$ by $6 x^{2} e^{x y}$ and simplify.

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Step 5.3.1

Divide each term in $6 x^{2} y' e^{x y} = - 6 x y e^{x y} - 6 e^{x y}$ by $6 x^{2} e^{x y}$ .

$\frac{6 x^{2} y' e^{x y}}{6 x^{2} e^{x y}} = \frac{- 6 x y e^{x y}}{6 x^{2} e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $6$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{6 x^{2} y' e^{x y}}{6 x^{2} e^{x y}} = \frac{- 6 x y e^{x y}}{6 x^{2} e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{x^{2} y' e^{x y}}{x^{2} e^{x y}} = \frac{- 6 x y e^{x y}}{6 x^{2} e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.2.2

Cancel the common factor of $x^{2}$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{x^{2} y' e^{x y}}{x^{2} e^{x y}} = \frac{- 6 x y e^{x y}}{6 x^{2} e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.2.2.2

Rewrite the expression.

$\frac{y' e^{x y}}{e^{x y}} = \frac{- 6 x y e^{x y}}{6 x^{2} e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.2.3

Cancel the common factor of $e^{x y}$ .

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Step 5.3.2.3.1

Cancel the common factor.

$\frac{y' e^{x y}}{e^{x y}} = \frac{- 6 x y e^{x y}}{6 x^{2} e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{- 6 x y e^{x y}}{6 x^{2} e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

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

Cancel the common factor of $- 6$ and $6$ .

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Step 5.3.3.1.1.1

Factor $6$ out of $- 6 x y e^{x y}$ .

$y' = \frac{6 (- x y e^{x y})}{6 x^{2} e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Factor $6$ out of $6 x^{2} e^{x y}$ .

$y' = \frac{6 (- x y e^{x y})}{6 (x^{2} e^{x y})} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.1.2.2

Cancel the common factor.

$y' = \frac{6 (- x y e^{x y})}{6 (x^{2} e^{x y})} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- x y e^{x y}}{x^{2} e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.2

Cancel the common factor of $x$ and $x^{2}$ .

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Step 5.3.3.1.2.1

Factor $x$ out of $- x y e^{x y}$ .

$y' = \frac{x (- y e^{x y})}{x^{2} e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.2.2

Cancel the common factors.

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Step 5.3.3.1.2.2.1

Factor $x$ out of $x^{2} e^{x y}$ .

$y' = \frac{x (- y e^{x y})}{x (x e^{x y})} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.2.2.2

Cancel the common factor.

$y' = \frac{x (- y e^{x y})}{x (x e^{x y})} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.2.2.3

Rewrite the expression.

$y' = \frac{- y e^{x y}}{x e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.3

Cancel the common factor of $e^{x y}$ .

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Step 5.3.3.1.3.1

Cancel the common factor.

$y' = \frac{- y e^{x y}}{x e^{x y}} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.3.2

Rewrite the expression.

$y' = \frac{- y}{x} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{y}{x} + \frac{- 6 e^{x y}}{6 x^{2} e^{x y}}$

Step 5.3.3.1.5

Cancel the common factor of $- 6$ and $6$ .

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Step 5.3.3.1.5.1

Factor $6$ out of $- 6 e^{x y}$ .

$y' = - \frac{y}{x} + \frac{6 (- e^{x y})}{6 x^{2} e^{x y}}$

Step 5.3.3.1.5.2

Cancel the common factors.

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Step 5.3.3.1.5.2.1

Factor $6$ out of $6 x^{2} e^{x y}$ .

$y' = - \frac{y}{x} + \frac{6 (- e^{x y})}{6 (x^{2} e^{x y})}$

Step 5.3.3.1.5.2.2

Cancel the common factor.

$y' = - \frac{y}{x} + \frac{6 (- e^{x y})}{6 (x^{2} e^{x y})}$

Step 5.3.3.1.5.2.3

Rewrite the expression.

$y' = - \frac{y}{x} + \frac{- e^{x y}}{x^{2} e^{x y}}$

Step 5.3.3.1.6

Cancel the common factor of $e^{x y}$ .

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Step 5.3.3.1.6.1

Cancel the common factor.

$y' = - \frac{y}{x} + \frac{- e^{x y}}{x^{2} e^{x y}}$

Step 5.3.3.1.6.2

Rewrite the expression.

$y' = - \frac{y}{x} + \frac{- 1}{x^{2}}$

Step 5.3.3.1.7

Move the negative in front of the fraction.

$y' = - \frac{y}{x} - \frac{1}{x^{2}}$

Step 6

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

$\frac{d y}{d x} = - \frac{y}{x} - \frac{1}{x^{2}}$