Calculus Examples

$2 x^{2} + 6 \ln (x y) = 15$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (2 x^{2} + 6 \ln (x y)) = \frac{d}{d x} (15)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $2 x^{2} + 6 \ln (x y)$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [6 \ln (x y)]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [6 \ln (x y)]$

Step 2.2

Evaluate $\frac{d}{d x} [2 x^{2}]$ .

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Step 2.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 \ln (x y)]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 (2 x) + \frac{d}{d x} [6 \ln (x y)]$

Step 2.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [6 \ln (x y)]$

Step 2.3

Evaluate $\frac{d}{d x} [6 \ln (x y)]$ .

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Step 2.3.1

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

$4 x + 6 \frac{d}{d x} [\ln (x y)]$

Step 2.3.2

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

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Step 2.3.2.1

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

$4 x + 6 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x y])$

Step 2.3.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$4 x + 6 (\frac{1}{u} \frac{d}{d x} [x y])$

Step 2.3.2.3

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

$4 x + 6 (\frac{1}{x y} \frac{d}{d x} [x y])$

Step 2.3.3

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 $g (x) = y$ .

$4 x + 6 (\frac{1}{x y} (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]))$

Step 2.3.4

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

$4 x + 6 (\frac{1}{x y} (x y' + y \frac{d}{d x} [x]))$

Step 2.3.5

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

$4 x + 6 (\frac{1}{x y} (x y' + y \cdot 1))$

Step 2.3.6

Multiply $y$ by $1$ .

$4 x + 6 (\frac{1}{x y} (x y' + y))$

Step 2.3.7

Combine $\frac{1}{x y}$ and $6$ .

$4 x + \frac{6}{x y} (x y' + y)$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$4 x + \frac{6}{x y} (x y') + \frac{6}{x y} y$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $x$ and $\frac{6}{x y}$ .

$4 x + \frac{x \cdot 6}{x y} y' + \frac{6}{x y} y$

Step 2.4.2.2

Combine $\frac{x \cdot 6}{x y}$ and $y'$ .

$4 x + \frac{x \cdot 6 y'}{x y} + \frac{6}{x y} y$

Step 2.4.2.3

Move $6$ to the left of $x$ .

$4 x + \frac{6 \cdot x y'}{x y} + \frac{6}{x y} y$

Step 2.4.2.4

Cancel the common factor of $x$ .

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Step 2.4.2.4.1

Cancel the common factor.

$4 x + \frac{6 x y'}{x y} + \frac{6}{x y} y$

Step 2.4.2.4.2

Rewrite the expression.

$4 x + \frac{6 y'}{y} + \frac{6}{x y} y$

Step 2.4.2.5

Combine $\frac{6}{x y}$ and $y$ .

$4 x + \frac{6 y'}{y} + \frac{6 y}{x y}$

Step 2.4.2.6

Cancel the common factor of $y$ .

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Step 2.4.2.6.1

Cancel the common factor.

$4 x + \frac{6 y'}{y} + \frac{6 y}{x y}$

Step 2.4.2.6.2

Rewrite the expression.

$4 x + \frac{6 y'}{y} + \frac{6}{x}$

Step 3

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$0$

Step 4

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

$4 x + \frac{6 y'}{y} + \frac{6}{x} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Subtract $4 x$ from both sides of the equation.

$\frac{6 y'}{y} + \frac{6}{x} = - 4 x$

Step 5.1.2

Subtract $\frac{6}{x}$ from both sides of the equation.

$\frac{6 y'}{y} = - 4 x - \frac{6}{x}$

Step 5.2

Multiply both sides by $y$ .

$\frac{6 y'}{y} y = (- 4 x - \frac{6}{x}) y$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $y$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{6 y'}{y} y = (- 4 x - \frac{6}{x}) y$

Step 5.3.1.1.2

Rewrite the expression.

$6 y' = (- 4 x - \frac{6}{x}) y$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $(- 4 x - \frac{6}{x}) y$ .

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Step 5.3.2.1.1

Simplify terms.

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Step 5.3.2.1.1.1

Apply the distributive property.

$6 y' = - 4 x y - \frac{6}{x} y$

Step 5.3.2.1.1.2

Combine $y$ and $\frac{6}{x}$ .

$6 y' = - 4 x y - \frac{y \cdot 6}{x}$

Step 5.3.2.1.2

Move $6$ to the left of $y$ .

$6 y' = - 4 x y - \frac{6 y}{x}$

Step 5.4

Divide each term in $6 y' = - 4 x y - \frac{6 y}{x}$ by $6$ and simplify.

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Step 5.4.1

Divide each term in $6 y' = - 4 x y - \frac{6 y}{x}$ by $6$ .

$\frac{6 y'}{6} = \frac{- 4 x y}{6} + \frac{- \frac{6 y}{x}}{6}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $6$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{6 y'}{6} = \frac{- 4 x y}{6} + \frac{- \frac{6 y}{x}}{6}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 4 x y}{6} + \frac{- \frac{6 y}{x}}{6}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Cancel the common factor of $- 4$ and $6$ .

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Step 5.4.3.1.1.1

Factor $2$ out of $- 4 x y$ .

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

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Factor $2$ out of $6$ .

$y' = \frac{2 (- 2 x y)}{2 (3)} + \frac{- \frac{6 y}{x}}{6}$

Step 5.4.3.1.1.2.2

Cancel the common factor.

$y' = \frac{2 (- 2 x y)}{2 \cdot 3} + \frac{- \frac{6 y}{x}}{6}$

Step 5.4.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- 2 x y}{3} + \frac{- \frac{6 y}{x}}{6}$

Step 5.4.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{2 x y}{3} + \frac{- \frac{6 y}{x}}{6}$

Step 5.4.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{2 x y}{3} - \frac{6 y}{x} \cdot \frac{1}{6}$

Step 5.4.3.1.4

Cancel the common factor of $6$ .

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Step 5.4.3.1.4.1

Move the leading negative in $- \frac{6 y}{x}$ into the numerator.

$y' = - \frac{2 x y}{3} + \frac{- 6 y}{x} \cdot \frac{1}{6}$

Step 5.4.3.1.4.2

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

$y' = - \frac{2 x y}{3} + \frac{6 (- y)}{x} \cdot \frac{1}{6}$

Step 5.4.3.1.4.3

Cancel the common factor.

$y' = - \frac{2 x y}{3} + \frac{6 (- y)}{x} \cdot \frac{1}{6}$

Step 5.4.3.1.4.4

Rewrite the expression.

$y' = - \frac{2 x y}{3} + \frac{- y}{x}$

Step 5.4.3.1.5

Move the negative in front of the fraction.

$y' = - \frac{2 x y}{3} - \frac{y}{x}$

Step 6

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

$\frac{d y}{d x} = - \frac{2 x y}{3} - \frac{y}{x}$