Calculus Examples

$3 x y = \ln (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 x y) = \frac{d}{d x} (\ln (x))$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x y$ with respect to $x$ is $3 \frac{d}{d x} [x y]$ .

$3 \frac{d}{d x} [x y]$

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 $g (x) = y$ .

$3 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 2.3

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

$3 (x y' + y \frac{d}{d x} [x])$

Step 2.4

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

$3 (x y' + y \cdot 1)$

Step 2.5

Multiply $y$ by $1$ .

$3 (x y' + y)$

Step 2.6

Apply the distributive property.

$3 x y' + 3 y$

Step 3

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

$\frac{1}{x}$

Step 4

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

$3 x y' + 3 y = \frac{1}{x}$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $3 y$ from both sides of the equation.

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

Step 5.2

Divide each term in $3 x y' = \frac{1}{x} - 3 y$ by $3 x$ and simplify.

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Step 5.2.1

Divide each term in $3 x y' = \frac{1}{x} - 3 y$ by $3 x$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

Cancel the common factor of $x$ .

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.3.1.2

Multiply $\frac{1}{x} \cdot \frac{1}{3 x}$ .

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Step 5.2.3.1.2.1

Multiply $\frac{1}{x}$ by $\frac{1}{3 x}$ .

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

Step 5.2.3.1.2.2

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

$y' = \frac{1}{3 (x^{1} x)} + \frac{- 3 y}{3 x}$

Step 5.2.3.1.2.3

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

$y' = \frac{1}{3 (x^{1} x^{1})} + \frac{- 3 y}{3 x}$

Step 5.2.3.1.2.4

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

$y' = \frac{1}{3 x^{1 + 1}} + \frac{- 3 y}{3 x}$

Step 5.2.3.1.2.5

Add $1$ and $1$ .

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

Step 5.2.3.1.3

Cancel the common factor of $- 3$ and $3$ .

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Step 5.2.3.1.3.1

Factor $3$ out of $- 3 y$ .

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

Step 5.2.3.1.3.2

Cancel the common factors.

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Step 5.2.3.1.3.2.1

Factor $3$ out of $3 x$ .

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

Step 5.2.3.1.3.2.2

Cancel the common factor.

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

Step 5.2.3.1.3.2.3

Rewrite the expression.

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

Step 5.2.3.1.4

Move the negative in front of the fraction.

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

Step 6

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

$\frac{d y}{d x} = \frac{1}{3 x^{2}} - \frac{y}{x}$

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Find the LCD of the terms in the equation.

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Step 7.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 x^{2}, x, 1$

Step 7.1.2

Since $3 x^{2}, x, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $3, 1, 1$ then find LCM for the variable part $x^{2}, x^{1}$ .

Step 7.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 7.1.4

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 7.1.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 7.1.6

The LCM of $3, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 7.1.7

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 7.1.8

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 7.1.9

The LCM of $x^{2}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x$

Step 7.1.10

Multiply $x$ by $x$ .

$x^{2}$

Step 7.1.11

The LCM for $3 x^{2}, x, 1$ is the numeric part $3$ multiplied by the variable part.

$3 x^{2}$

Step 7.2

Multiply each term in $\frac{1}{3 x^{2}} - \frac{y}{x} = 0$ by $3 x^{2}$ to eliminate the fractions.

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Step 7.2.1

Multiply each term in $\frac{1}{3 x^{2}} - \frac{y}{x} = 0$ by $3 x^{2}$ .

$\frac{1}{3 x^{2}} (3 x^{2}) - \frac{y}{x} (3 x^{2}) = 0 (3 x^{2})$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Simplify each term.

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Step 7.2.2.1.1

Rewrite using the commutative property of multiplication.

$3 \frac{1}{3 x^{2}} x^{2} - \frac{y}{x} (3 x^{2}) = 0 (3 x^{2})$

Step 7.2.2.1.2

Cancel the common factor of $3$ .

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Step 7.2.2.1.2.1

Factor $3$ out of $3 x^{2}$ .

$3 \frac{1}{3 (x^{2})} x^{2} - \frac{y}{x} (3 x^{2}) = 0 (3 x^{2})$

Step 7.2.2.1.2.2

Cancel the common factor.

$3 \frac{1}{3 x^{2}} x^{2} - \frac{y}{x} (3 x^{2}) = 0 (3 x^{2})$

Step 7.2.2.1.2.3

Rewrite the expression.

$\frac{1}{x^{2}} x^{2} - \frac{y}{x} (3 x^{2}) = 0 (3 x^{2})$

Step 7.2.2.1.3

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

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Step 7.2.2.1.3.1

Cancel the common factor.

$\frac{1}{x^{2}} x^{2} - \frac{y}{x} (3 x^{2}) = 0 (3 x^{2})$

Step 7.2.2.1.3.2

Rewrite the expression.

$1 - \frac{y}{x} (3 x^{2}) = 0 (3 x^{2})$

Step 7.2.2.1.4

Cancel the common factor of $x$ .

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Step 7.2.2.1.4.1

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

$1 + \frac{- y}{x} (3 x^{2}) = 0 (3 x^{2})$

Step 7.2.2.1.4.2

Factor $x$ out of $3 x^{2}$ .

$1 + \frac{- y}{x} (x (3 x)) = 0 (3 x^{2})$

Step 7.2.2.1.4.3

Cancel the common factor.

$1 + \frac{- y}{x} (x (3 x)) = 0 (3 x^{2})$

Step 7.2.2.1.4.4

Rewrite the expression.

$1 - y (3 x) = 0 (3 x^{2})$

Step 7.2.2.1.5

Multiply $3$ by $- 1$ .

$1 - 3 y x = 0 (3 x^{2})$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Multiply $0 (3 x^{2})$ .

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Step 7.2.3.1.1

Multiply $3$ by $0$ .

$1 - 3 y x = 0 x^{2}$

Step 7.2.3.1.2

Multiply $0$ by $x^{2}$ .

$1 - 3 y x = 0$

Step 7.3

Solve the equation.

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Step 7.3.1

Subtract $1$ from both sides of the equation.

$- 3 y x = - 1$

Step 7.3.2

Divide each term in $- 3 y x = - 1$ by $- 3 y$ and simplify.

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Step 7.3.2.1

Divide each term in $- 3 y x = - 1$ by $- 3 y$ .

$\frac{- 3 y x}{- 3 y} = \frac{- 1}{- 3 y}$

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

Cancel the common factor of $- 3$ .

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Step 7.3.2.2.1.1

Cancel the common factor.

$\frac{- 3 y x}{- 3 y} = \frac{- 1}{- 3 y}$

Step 7.3.2.2.1.2

Rewrite the expression.

$\frac{y x}{y} = \frac{- 1}{- 3 y}$

Step 7.3.2.2.2

Cancel the common factor of $y$ .

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Step 7.3.2.2.2.1

Cancel the common factor.

$\frac{y x}{y} = \frac{- 1}{- 3 y}$

Step 7.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 3 y}$

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{3 y}$

Step 8

Solve for $y$ .

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Step 8.1

Simplify $3 (\frac{1}{3 y}) \cdot y$ .

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Step 8.1.1

Rewrite.

$0 + 0 + 3 (\frac{1}{3 y}) \cdot y = \ln (\frac{1}{3 y})$

Step 8.1.2

Simplify by adding zeros.

$3 (\frac{1}{3 y}) \cdot y = \ln (\frac{1}{3 y})$

Step 8.1.3

Cancel the common factor of $3$ .

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Step 8.1.3.1

Factor $3$ out of $3 y$ .

$3 \frac{1}{3 (y)} \cdot y = \ln (\frac{1}{3 y})$

Step 8.1.3.2

Cancel the common factor.

$3 \frac{1}{3 y} \cdot y = \ln (\frac{1}{3 y})$

Step 8.1.3.3

Rewrite the expression.

$\frac{1}{y} \cdot y = \ln (\frac{1}{3 y})$

Step 8.1.4

Cancel the common factor of $y$ .

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Step 8.1.4.1

Cancel the common factor.

$\frac{1}{y} \cdot y = \ln (\frac{1}{3 y})$

Step 8.1.4.2

Rewrite the expression.

$1 = \ln (\frac{1}{3 y})$

Step 8.2

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\ln (\frac{1}{3 y}) = 1$

Step 8.3

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{1}{3 y})} = e^{1}$

Step 8.4

Rewrite $\ln (\frac{1}{3 y}) = 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{1} = \frac{1}{3 y}$

Step 8.5

Solve for $y$ .

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Step 8.5.1

Rewrite the equation as $\frac{1}{3 y} = e^{1}$ .

$\frac{1}{3 y} = e$

Step 8.5.2

Find the LCD of the terms in the equation.

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Step 8.5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 y, 1$

Step 8.5.2.2

The LCM of one and any expression is the expression.

$3 y$

Step 8.5.3

Multiply each term in $\frac{1}{3 y} = e$ by $3 y$ to eliminate the fractions.

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Step 8.5.3.1

Multiply each term in $\frac{1}{3 y} = e$ by $3 y$ .

$\frac{1}{3 y} (3 y) = e (3 y)$

Step 8.5.3.2

Simplify the left side.

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Step 8.5.3.2.1

Rewrite using the commutative property of multiplication.

$3 \frac{1}{3 y} y = e (3 y)$

Step 8.5.3.2.2

Cancel the common factor of $3$ .

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Step 8.5.3.2.2.1

Factor $3$ out of $3 y$ .

$3 \frac{1}{3 (y)} y = e (3 y)$

Step 8.5.3.2.2.2

Cancel the common factor.

$3 \frac{1}{3 y} y = e (3 y)$

Step 8.5.3.2.2.3

Rewrite the expression.

$\frac{1}{y} y = e (3 y)$

Step 8.5.3.2.3

Cancel the common factor of $y$ .

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Step 8.5.3.2.3.1

Cancel the common factor.

$\frac{1}{y} y = e (3 y)$

Step 8.5.3.2.3.2

Rewrite the expression.

$1 = e (3 y)$

Step 8.5.3.3

Simplify the right side.

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Step 8.5.3.3.1

Move $3$ to the left of $e$ .

$1 = 3 e y$

Step 8.5.4

Solve the equation.

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Step 8.5.4.1

Rewrite the equation as $3 e y = 1$ .

$3 e y = 1$

Step 8.5.4.2

Divide each term in $3 e y = 1$ by $3 e$ and simplify.

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Step 8.5.4.2.1

Divide each term in $3 e y = 1$ by $3 e$ .

$\frac{3 e y}{3 e} = \frac{1}{3 e}$

Step 8.5.4.2.2

Simplify the left side.

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Step 8.5.4.2.2.1

Cancel the common factor of $3$ .

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Step 8.5.4.2.2.1.1

Cancel the common factor.

$\frac{3 e y}{3 e} = \frac{1}{3 e}$

Step 8.5.4.2.2.1.2

Rewrite the expression.

$\frac{e y}{e} = \frac{1}{3 e}$

Step 8.5.4.2.2.2

Cancel the common factor of $e$ .

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Step 8.5.4.2.2.2.1

Cancel the common factor.

$\frac{e y}{e} = \frac{1}{3 e}$

Step 8.5.4.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{1}{3 e}$

Step 9

Solve for $x$ when $y$ is $\frac{1}{3 e}$ .

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Step 9.1

Multiply $3$ by $\frac{1}{3 e}$ .

$x = \frac{1}{3 (\frac{1}{3 e})}$

Step 9.2

Simplify $\frac{1}{3 (\frac{1}{3 e})}$ .

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Step 9.2.1

Combine $3$ and $\frac{1}{3 e}$ .

$x = \frac{1}{\frac{3}{3 e}}$

Step 9.2.2

Reduce the expression $\frac{3}{3 e}$ by cancelling the common factors.

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Step 9.2.2.1

Cancel the common factor.

$x = \frac{1}{\frac{3}{3 e}}$

Step 9.2.2.2

Rewrite the expression.

$x = \frac{1}{\frac{1}{e}}$

Step 9.2.3

Multiply the numerator by the reciprocal of the denominator.

$x = 1 e$

Step 9.2.4

Multiply $e$ by $1$ .

$x = e$

Step 10

Find the points where $\frac{d y}{d x} = 0$ .

$(e, \frac{1}{3 e})$

Step 11