Calculus Examples

$\ln (x y) + 8 x = 45$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\ln (x y) + 8 x) = \frac{d}{d x} (45)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $\ln (x y) + 8 x$ with respect to $x$ is $\frac{d}{d x} [\ln (x y)] + \frac{d}{d x} [8 x]$ .

$\frac{d}{d x} [\ln (x y)] + \frac{d}{d x} [8 x]$

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

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

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x y] + \frac{d}{d x} [8 x]$

Step 2.2.1.2

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

$\frac{1}{u} \frac{d}{d x} [x y] + \frac{d}{d x} [8 x]$

Step 2.2.1.3

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

$\frac{1}{x y} \frac{d}{d x} [x y] + \frac{d}{d x} [8 x]$

Step 2.2.2

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 $g (x) = y$ .

$\frac{1}{x y} (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [8 x]$

Step 2.2.3

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

$\frac{1}{x y} (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [8 x]$

Step 2.2.4

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

$\frac{1}{x y} (x y' + y \cdot 1) + \frac{d}{d x} [8 x]$

Step 2.2.5

Multiply $y$ by $1$ .

$\frac{1}{x y} (x y' + y) + \frac{d}{d x} [8 x]$

Step 2.3

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 2.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\frac{1}{x y} (x y' + y) + 8 \frac{d}{d x} [x]$

Step 2.3.2

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

$\frac{1}{x y} (x y' + y) + 8 \cdot 1$

Step 2.3.3

Multiply $8$ by $1$ .

$\frac{1}{x y} (x y' + y) + 8$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$\frac{1}{x y} (x y') + \frac{1}{x y} y + 8$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

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

$\frac{x}{x y} y' + \frac{1}{x y} y + 8$

Step 2.4.2.2

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

$\frac{x y'}{x y} + \frac{1}{x y} y + 8$

Step 2.4.2.3

Cancel the common factor of $x$ .

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Step 2.4.2.3.1

Cancel the common factor.

$\frac{x y'}{x y} + \frac{1}{x y} y + 8$

Step 2.4.2.3.2

Rewrite the expression.

$\frac{y'}{y} + \frac{1}{x y} y + 8$

Step 2.4.2.4

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

$\frac{y'}{y} + \frac{y}{x y} + 8$

Step 2.4.2.5

Cancel the common factor of $y$ .

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Step 2.4.2.5.1

Cancel the common factor.

$\frac{y'}{y} + \frac{y}{x y} + 8$

Step 2.4.2.5.2

Rewrite the expression.

$\frac{y'}{y} + \frac{1}{x} + 8$

Step 3

Since $45$ is constant with respect to $x$ , the derivative of $45$ with respect to $x$ is $0$ .

$0$

Step 4

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

$\frac{y'}{y} + \frac{1}{x} + 8 = 0$

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 $\frac{1}{x}$ from both sides of the equation.

$\frac{y'}{y} + 8 = - \frac{1}{x}$

Step 5.1.2

Subtract $8$ from both sides of the equation.

$\frac{y'}{y} = - \frac{1}{x} - 8$

Step 5.2

Multiply both sides by $y$ .

$\frac{y'}{y} y = (- \frac{1}{x} - 8) y$

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.

$\frac{y'}{y} y = (- \frac{1}{x} - 8) y$

Step 5.3.1.1.2

Rewrite the expression.

$y' = (- \frac{1}{x} - 8) y$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $(- \frac{1}{x} - 8) y$ .

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Step 5.3.2.1.1

Apply the distributive property.

$y' = - \frac{1}{x} y - 8 y$

Step 5.3.2.1.2

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

$y' = - \frac{y}{x} - 8 y$

Step 6

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

$\frac{d y}{d x} = - \frac{y}{x} - 8 y$