Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 5 x y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $5 x y$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [5 x y]$

Step 2.1.2

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

$e^{u} \frac{d}{d x} [5 x y]$

Step 2.1.3

Replace all occurrences of $u$ with $5 x y$ .

$e^{5 x y} \frac{d}{d x} [5 x y]$

Step 2.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x y$ with respect to $x$ is $5 \frac{d}{d x} [x y]$ .

$e^{5 x y} (5 \frac{d}{d x} [x y])$

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 $g (x) = y$ .

$e^{5 x y} (5 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]))$

Step 2.4

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

$e^{5 x y} (5 (x y' + y \frac{d}{d x} [x]))$

Step 2.5

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

$e^{5 x y} (5 (x y' + y \cdot 1))$

Step 2.6

Multiply $y$ by $1$ .

$e^{5 x y} (5 (x y' + y))$

Step 2.7

Simplify.

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Step 2.7.1

Apply the distributive property.

$e^{5 x y} (5 (x y') + 5 y)$

Step 2.7.2

Apply the distributive property.

$e^{5 x y} (5 (x y')) + e^{5 x y} (5 y)$

Step 2.7.3

Reorder terms.

$5 e^{5 x y} x y' + 5 e^{5 x y} y$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

$6 y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Reorder factors in $5 e^{5 x y} x y' + 5 e^{5 x y} y$ .

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

Step 5.2

Subtract $6 y'$ from both sides of the equation.

$5 x y' e^{5 x y} + 5 y e^{5 x y} - 6 y' = 0$

Step 5.3

Subtract $5 y e^{5 x y}$ from both sides of the equation.

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

Step 5.4

Factor $y'$ out of $5 x y' e^{5 x y} - 6 y'$ .

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Step 5.4.1

Factor $y'$ out of $5 x y' e^{5 x y}$ .

$y' (5 x e^{5 x y}) - 6 y' = - 5 y e^{5 x y}$

Step 5.4.2

Factor $y'$ out of $- 6 y'$ .

$y' (5 x e^{5 x y}) + y' \cdot - 6 = - 5 y e^{5 x y}$

Step 5.4.3

Factor $y'$ out of $y' (5 x e^{5 x y}) + y' \cdot - 6$ .

$y' (5 x e^{5 x y} - 6) = - 5 y e^{5 x y}$

Step 5.5

Divide each term in $y' (5 x e^{5 x y} - 6) = - 5 y e^{5 x y}$ by $5 x e^{5 x y} - 6$ and simplify.

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Step 5.5.1

Divide each term in $y' (5 x e^{5 x y} - 6) = - 5 y e^{5 x y}$ by $5 x e^{5 x y} - 6$ .

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $5 x e^{5 x y} - 6$ .

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 5 y e^{5 x y}}{5 x e^{5 x y} - 6}$

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.

$y' = - \frac{5 y e^{5 x y}}{5 x e^{5 x y} - 6}$

Step 6

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

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