Algebra Examples

$x \ln (y) + 5 y = 2 x^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x \ln (y) + 5 y) = \frac{d}{d y} (2 x^{3})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $x \ln (y) + 5 y$ with respect to $y$ is $\frac{d}{d y} [x \ln (y)] + \frac{d}{d y} [5 y]$ .

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

Step 2.2

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

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = \ln (y)$ .

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

Step 2.2.2

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

$x \frac{1}{y} + \ln (y) \frac{d}{d y} [x] + \frac{d}{d y} [5 y]$

Step 2.2.3

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

$x \frac{1}{y} + \ln (y) x' + \frac{d}{d y} [5 y]$

Step 2.2.4

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

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

Step 2.3

Evaluate $\frac{d}{d y} [5 y]$ .

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Step 2.3.1

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

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

Step 2.3.2

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

$\frac{x}{y} + \ln (y) x' + 5 \cdot 1$

Step 2.3.3

Multiply $5$ by $1$ .

$\frac{x}{y} + \ln (y) x' + 5$

Step 2.4

Reorder terms.

$\ln (y) x' + \frac{x}{y} + 5$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $2$ is constant with respect to $y$ , the derivative of $2 x^{3}$ with respect to $y$ is $2 \frac{d}{d y} [x^{3}]$ .

$2 \frac{d}{d y} [x^{3}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{3}$ and $g (y) = x$ .

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Step 3.2.1

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

$2 (\frac{d}{d u} [u^{3}] \frac{d}{d y} [x])$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$2 (3 u^{2} \frac{d}{d y} [x])$

Step 3.2.3

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

$2 (3 x^{2} \frac{d}{d y} [x])$

Step 3.3

Multiply $3$ by $2$ .

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

Step 3.4

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

$6 x^{2} x'$

Step 4

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

$\ln (y) x' + \frac{x}{y} + 5 = 6 x^{2} x'$

Step 5

Solve for $x'$ .

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Step 5.1

Reorder factors in $\ln (y) x' + \frac{x}{y} + 5$ .

$x' \ln (y) + \frac{x}{y} + 5 = 6 x^{2} x'$

Step 5.2

Subtract $6 x^{2} x'$ from both sides of the equation.

$x' \ln (y) + \frac{x}{y} + 5 - 6 x^{2} x' = 0$

Step 5.3

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

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Step 5.3.1

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

$x' \ln (y) + 5 - 6 x^{2} x' = - \frac{x}{y}$

Step 5.3.2

Subtract $5$ from both sides of the equation.

$x' \ln (y) - 6 x^{2} x' = - \frac{x}{y} - 5$

Step 5.4

Factor $x'$ out of $x' \ln (y) - 6 x^{2} x'$ .

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Step 5.4.1

Factor $x'$ out of $x' \ln (y)$ .

$x' (\ln (y)) - 6 x^{2} x' = - \frac{x}{y} - 5$

Step 5.4.2

Factor $x'$ out of $- 6 x^{2} x'$ .

$x' (\ln (y)) + x' (- 6 x^{2}) = - \frac{x}{y} - 5$

Step 5.4.3

Factor $x'$ out of $x' (\ln (y)) + x' (- 6 x^{2})$ .

$x' (\ln (y) - 6 x^{2}) = - \frac{x}{y} - 5$

Step 5.5

Divide each term in $x' (\ln (y) - 6 x^{2}) = - \frac{x}{y} - 5$ by $\ln (y) - 6 x^{2}$ and simplify.

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Step 5.5.1

Divide each term in $x' (\ln (y) - 6 x^{2}) = - \frac{x}{y} - 5$ by $\ln (y) - 6 x^{2}$ .

$\frac{x' (\ln (y) - 6 x^{2})}{\ln (y) - 6 x^{2}} = \frac{- \frac{x}{y}}{\ln (y) - 6 x^{2}} + \frac{- 5}{\ln (y) - 6 x^{2}}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $\ln (y) - 6 x^{2}$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{x' (\ln (y) - 6 x^{2})}{\ln (y) - 6 x^{2}} = \frac{- \frac{x}{y}}{\ln (y) - 6 x^{2}} + \frac{- 5}{\ln (y) - 6 x^{2}}$

Step 5.5.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{- \frac{x}{y}}{\ln (y) - 6 x^{2}} + \frac{- 5}{\ln (y) - 6 x^{2}}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Combine the numerators over the common denominator.

$x' = \frac{- \frac{x}{y} - 5}{\ln (y) - 6 x^{2}}$

Step 5.5.3.2

Multiply the numerator and denominator of the fraction by $y$ .

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Step 5.5.3.2.1

Multiply $\frac{- \frac{x}{y} - 5}{\ln (y) - 6 x^{2}}$ by $\frac{y}{y}$ .

$x' = \frac{y}{y} \cdot \frac{- \frac{x}{y} - 5}{\ln (y) - 6 x^{2}}$

Step 5.5.3.2.2

Combine.

$x' = \frac{y (- \frac{x}{y} - 5)}{y (\ln (y) - 6 x^{2})}$

Step 5.5.3.3

Apply the distributive property.

$x' = \frac{y (- \frac{x}{y}) + y \cdot - 5}{y \ln (y) + y (- 6 x^{2})}$

Step 5.5.3.4

Cancel the common factor of $y$ .

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Step 5.5.3.4.1

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

$x' = \frac{y \frac{- x}{y} + y \cdot - 5}{y \ln (y) + y (- 6 x^{2})}$

Step 5.5.3.4.2

Cancel the common factor.

$x' = \frac{y \frac{- x}{y} + y \cdot - 5}{y \ln (y) + y (- 6 x^{2})}$

Step 5.5.3.4.3

Rewrite the expression.

$x' = \frac{- x + y \cdot - 5}{y \ln (y) + y (- 6 x^{2})}$

Step 5.5.3.5

Move $- 5$ to the left of $y$ .

$x' = \frac{- x - 5 y}{y \ln (y) + y (- 6 x^{2})}$

Step 5.5.3.6

Simplify with factoring out.

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Step 5.5.3.6.1

Factor $y$ out of $y \ln (y) + y (- 6 x^{2})$ .

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Step 5.5.3.6.1.1

Factor $y$ out of $y \ln (y)$ .

$x' = \frac{- x - 5 y}{y (\ln (y)) + y (- 6 x^{2})}$

Step 5.5.3.6.1.2

Factor $y$ out of $y (\ln (y)) + y (- 6 x^{2})$ .

$x' = \frac{- x - 5 y}{y (\ln (y) - 6 x^{2})}$

Step 5.5.3.6.2

Factor $- 1$ out of $- x$ .

$x' = \frac{- (x) - 5 y}{y (\ln (y) - 6 x^{2})}$

Step 5.5.3.6.3

Factor $- 1$ out of $- 5 y$ .

$x' = \frac{- (x) - (5 y)}{y (\ln (y) - 6 x^{2})}$

Step 5.5.3.6.4

Factor $- 1$ out of $- (x) - (5 y)$ .

$x' = \frac{- (x + 5 y)}{y (\ln (y) - 6 x^{2})}$

Step 5.5.3.6.5

Simplify the expression.

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Step 5.5.3.6.5.1

Rewrite $- (x + 5 y)$ as $- 1 (x + 5 y)$ .

$x' = \frac{- 1 (x + 5 y)}{y (\ln (y) - 6 x^{2})}$

Step 5.5.3.6.5.2

Move the negative in front of the fraction.

$x' = - \frac{x + 5 y}{y (\ln (y) - 6 x^{2})}$

Step 6

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

$\frac{d x}{d y} = - \frac{x + 5 y}{y (\ln (y) - 6 x^{2})}$