Calculus Examples

$7 \ln (x^{2} y^{2}) = 6$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (7 \ln (x^{2} y^{2})) = \frac{d}{d x} (6)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 \ln (x^{2} y^{2})$ with respect to $x$ is $7 \frac{d}{d x} [\ln (x^{2} y^{2})]$ .

$7 \frac{d}{d x} [\ln (x^{2} y^{2})]$

Step 2.2

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 $g (x) = x^{2} y^{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} y^{2}$ .

$7 (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [x^{2} y^{2}])$

Step 2.2.2

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

$7 (\frac{1}{u_{1}} \frac{d}{d x} [x^{2} y^{2}])$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $x^{2} y^{2}$ .

$7 (\frac{1}{x^{2} y^{2}} \frac{d}{d x} [x^{2} y^{2}])$

Step 2.3

Combine $\frac{1}{x^{2} y^{2}}$ and $7$ .

$\frac{7}{x^{2} y^{2}} \frac{d}{d x} [x^{2} y^{2}]$

Step 2.4

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 $f (x) = x^{2}$ and $g (x) = y^{2}$ .

$\frac{7}{x^{2} y^{2}} (x^{2} \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x^{2}])$

Step 2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 2.5.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$\frac{7}{x^{2} y^{2}} (x^{2} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}])$

Step 2.5.2

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

$\frac{7}{x^{2} y^{2}} (x^{2} (2 u_{2} \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}])$

Step 2.5.3

Replace all occurrences of $u_{2}$ with $y$ .

$\frac{7}{x^{2} y^{2}} (x^{2} (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}])$

Step 2.6

Move $2$ to the left of $x^{2}$ .

$\frac{7}{x^{2} y^{2}} (2 \cdot x^{2} y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x^{2}])$

Step 2.7

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

$\frac{7}{x^{2} y^{2}} (2 x^{2} y y' + y^{2} \frac{d}{d x} [x^{2}])$

Step 2.8

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

$\frac{7}{x^{2} y^{2}} (2 x^{2} y y' + y^{2} (2 x))$

Step 2.9

Move $2$ to the left of $y^{2}$ .

$\frac{7}{x^{2} y^{2}} (2 x^{2} y y' + 2 \cdot y^{2} x)$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$\frac{7}{x^{2} y^{2}} (2 x^{2} y y') + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2

Combine terms.

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Step 2.10.2.1

Combine $2$ and $\frac{7}{x^{2} y^{2}}$ .

$\frac{2 \cdot 7}{x^{2} y^{2}} (x^{2} y y') + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.2

Multiply $2$ by $7$ .

$\frac{14}{x^{2} y^{2}} (x^{2} y y') + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.3

Combine $x^{2}$ and $\frac{14}{x^{2} y^{2}}$ .

$\frac{x^{2} \cdot 14}{x^{2} y^{2}} (y y') + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.4

Combine $y$ and $\frac{x^{2} \cdot 14}{x^{2} y^{2}}$ .

$\frac{y (x^{2} \cdot 14)}{x^{2} y^{2}} y' + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.5

Combine $\frac{y (x^{2} \cdot 14)}{x^{2} y^{2}}$ and $y'$ .

$\frac{y (x^{2} \cdot 14) y'}{x^{2} y^{2}} + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.6

Move $14$ to the left of $x^{2}$ .

$\frac{y (14 \cdot x^{2}) y'}{x^{2} y^{2}} + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.7

Move $14$ to the left of $y$ .

$\frac{14 \cdot y x^{2} y'}{x^{2} y^{2}} + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.8

Cancel the common factor of $y$ and $y^{2}$ .

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Step 2.10.2.8.1

Factor $y$ out of $14 y x^{2} y'$ .

$\frac{y (14 x^{2} y')}{x^{2} y^{2}} + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.8.2

Cancel the common factors.

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Step 2.10.2.8.2.1

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

$\frac{y (14 x^{2} y')}{y (x^{2} y)} + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.8.2.2

Cancel the common factor.

$\frac{y (14 x^{2} y')}{y (x^{2} y)} + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.8.2.3

Rewrite the expression.

$\frac{14 x^{2} y'}{x^{2} y} + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.9

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

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Step 2.10.2.9.1

Cancel the common factor.

$\frac{14 x^{2} y'}{x^{2} y} + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.9.2

Rewrite the expression.

$\frac{14 y'}{y} + \frac{7}{x^{2} y^{2}} (2 y^{2} x)$

Step 2.10.2.10

Combine $2$ and $\frac{7}{x^{2} y^{2}}$ .

$\frac{14 y'}{y} + \frac{2 \cdot 7}{x^{2} y^{2}} (y^{2} x)$

Step 2.10.2.11

Multiply $2$ by $7$ .

$\frac{14 y'}{y} + \frac{14}{x^{2} y^{2}} (y^{2} x)$

Step 2.10.2.12

Combine $y^{2}$ and $\frac{14}{x^{2} y^{2}}$ .

$\frac{14 y'}{y} + \frac{y^{2} \cdot 14}{x^{2} y^{2}} x$

Step 2.10.2.13

Combine $\frac{y^{2} \cdot 14}{x^{2} y^{2}}$ and $x$ .

$\frac{14 y'}{y} + \frac{y^{2} \cdot 14 x}{x^{2} y^{2}}$

Step 2.10.2.14

Move $14$ to the left of $y^{2}$ .

$\frac{14 y'}{y} + \frac{14 \cdot y^{2} x}{x^{2} y^{2}}$

Step 2.10.2.15

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

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Step 2.10.2.15.1

Cancel the common factor.

$\frac{14 y'}{y} + \frac{14 y^{2} x}{x^{2} y^{2}}$

Step 2.10.2.15.2

Rewrite the expression.

$\frac{14 y'}{y} + \frac{14 x}{x^{2}}$

Step 2.10.2.16

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

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Step 2.10.2.16.1

Factor $x$ out of $14 x$ .

$\frac{14 y'}{y} + \frac{x \cdot 14}{x^{2}}$

Step 2.10.2.16.2

Cancel the common factors.

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Step 2.10.2.16.2.1

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

$\frac{14 y'}{y} + \frac{x \cdot 14}{x \cdot x}$

Step 2.10.2.16.2.2

Cancel the common factor.

$\frac{14 y'}{y} + \frac{x \cdot 14}{x \cdot x}$

Step 2.10.2.16.2.3

Rewrite the expression.

$\frac{14 y'}{y} + \frac{14}{x}$

Step 3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0$

Step 4

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

$\frac{14 y'}{y} + \frac{14}{x} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $\frac{14}{x}$ from both sides of the equation.

$\frac{14 y'}{y} = - \frac{14}{x}$

Step 5.2

Multiply both sides by $y$ .

$\frac{14 y'}{y} y = - \frac{14}{x} y$

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.

$\frac{14 y'}{y} y = - \frac{14}{x} y$

Step 5.3.1.1.2

Rewrite the expression.

$14 y' = - \frac{14}{x} y$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $- \frac{14}{x} y$ .

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Step 5.3.2.1.1

Combine $y$ and $\frac{14}{x}$ .

$14 y' = - \frac{y \cdot 14}{x}$

Step 5.3.2.1.2

Move $14$ to the left of $y$ .

$14 y' = - \frac{14 y}{x}$

Step 5.4

Divide each term in $14 y' = - \frac{14 y}{x}$ by $14$ and simplify.

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Step 5.4.1

Divide each term in $14 y' = - \frac{14 y}{x}$ by $14$ .

$\frac{14 y'}{14} = \frac{- \frac{14 y}{x}}{14}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $14$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{14 y'}{14} = \frac{- \frac{14 y}{x}}{14}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- \frac{14 y}{x}}{14}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{14 y}{x} \cdot \frac{1}{14}$

Step 5.4.3.2

Cancel the common factor of $14$ .

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Step 5.4.3.2.1

Move the leading negative in $- \frac{14 y}{x}$ into the numerator.

$y' = \frac{- 14 y}{x} \cdot \frac{1}{14}$

Step 5.4.3.2.2

Factor $14$ out of $- 14 y$ .

$y' = \frac{14 (- y)}{x} \cdot \frac{1}{14}$

Step 5.4.3.2.3

Cancel the common factor.

$y' = \frac{14 (- y)}{x} \cdot \frac{1}{14}$

Step 5.4.3.2.4

Rewrite the expression.

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

Step 5.4.3.3

Move the negative in front of the fraction.

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

Step 6

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

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