Calculus Examples

$x = \ln (1 - 7 y^{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x) = \frac{d}{d x} (\ln (1 - 7 y^{2}))$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = 1 - 7 y^{2}$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $1 - 7 y^{2}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u_{1}$ with $1 - 7 y^{2}$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $1 - 7 y^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 7 y^{2}]$ .

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

Step 3.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{1 - 7 y^{2}} (0 + \frac{d}{d x} [- 7 y^{2}])$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [- 7 y^{2}]$ .

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

Step 3.2.4

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

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

Step 3.2.5

Combine fractions.

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Step 3.2.5.1

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

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

Step 3.2.5.2

Move the negative in front of the fraction.

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

Step 3.3

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 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

Combine fractions.

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Step 3.4.1

Multiply $2$ by $- 1$ .

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $- 2$ by $7$ .

$\frac{- 14}{1 - 7 y^{2}} (y \frac{d}{d x} [y])$

Step 3.4.4

Combine $y$ and $\frac{- 14}{1 - 7 y^{2}}$ .

$\frac{y \cdot - 14}{1 - 7 y^{2}} \frac{d}{d x} [y]$

Step 3.4.5

Simplify the expression.

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Step 3.4.5.1

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

$\frac{- 14 \cdot y}{1 - 7 y^{2}} \frac{d}{d x} [y]$

Step 3.4.5.2

Move the negative in front of the fraction.

$- \frac{14 y}{1 - 7 y^{2}} \frac{d}{d x} [y]$

Step 3.5

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

$- \frac{14 y}{1 - 7 y^{2}} y'$

Step 3.6

Combine $y'$ and $\frac{14 y}{1 - 7 y^{2}}$ .

$- \frac{y' (14 y)}{1 - 7 y^{2}}$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Rewrite using the commutative property of multiplication.

$- \frac{14 y' y}{1 - 7 y^{2}}$

Step 3.7.2

Reorder factors in $14 y' y$ .

$- \frac{14 y y'}{1 - 7 y^{2}}$

Step 4

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

$1 = - \frac{14 y y'}{1 - 7 y^{2}}$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $- \frac{14 y y'}{1 - 7 y^{2}} = 1$ .

$- \frac{14 y y'}{1 - 7 y^{2}} = 1$

Step 5.2

Divide each term in $- \frac{14 y y'}{1 - 7 y^{2}} = 1$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- \frac{14 y y'}{1 - 7 y^{2}} = 1$ by $- 1$ .

$\frac{- \frac{14 y y'}{1 - 7 y^{2}}}{- 1} = \frac{1}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{14 y y'}{1 - 7 y^{2}}}{1} = \frac{1}{- 1}$

Step 5.2.2.2

Divide $\frac{14 y y'}{1 - 7 y^{2}}$ by $1$ .

$\frac{14 y y'}{1 - 7 y^{2}} = \frac{1}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $1$ by $- 1$ .

$\frac{14 y y'}{1 - 7 y^{2}} = - 1$

Step 5.3

Multiply both sides by $1 - 7 y^{2}$ .

$\frac{14 y y'}{1 - 7 y^{2}} (1 - 7 y^{2}) = - (1 - 7 y^{2})$

Step 5.4

Simplify.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Cancel the common factor of $1 - 7 y^{2}$ .

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Step 5.4.1.1.1

Cancel the common factor.

$\frac{14 y y'}{1 - 7 y^{2}} (1 - 7 y^{2}) = - (1 - 7 y^{2})$

Step 5.4.1.1.2

Rewrite the expression.

$14 y y' = - (1 - 7 y^{2})$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $- (1 - 7 y^{2})$ .

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Step 5.4.2.1.1

Apply the distributive property.

$14 y y' = - 1 \cdot 1 - (- 7 y^{2})$

Step 5.4.2.1.2

Simplify the expression.

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Step 5.4.2.1.2.1

Multiply $- 1$ by $1$ .

$14 y y' = - 1 - (- 7 y^{2})$

Step 5.4.2.1.2.2

Multiply $- 7$ by $- 1$ .

$14 y y' = - 1 + 7 y^{2}$

Step 5.4.2.1.2.3

Reorder $- 1$ and $7 y^{2}$ .

$14 y y' = 7 y^{2} - 1$

Step 5.5

Divide each term in $14 y y' = 7 y^{2} - 1$ by $14 y$ and simplify.

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Step 5.5.1

Divide each term in $14 y y' = 7 y^{2} - 1$ by $14 y$ .

$\frac{14 y y'}{14 y} = \frac{7 y^{2}}{14 y} + \frac{- 1}{14 y}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $14$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{14 y y'}{14 y} = \frac{7 y^{2}}{14 y} + \frac{- 1}{14 y}$

Step 5.5.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{7 y^{2}}{14 y} + \frac{- 1}{14 y}$

Step 5.5.2.2

Cancel the common factor of $y$ .

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Step 5.5.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{7 y^{2}}{14 y} + \frac{- 1}{14 y}$

Step 5.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{7 y^{2}}{14 y} + \frac{- 1}{14 y}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Cancel the common factor of $7$ and $14$ .

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Step 5.5.3.1.1.1

Factor $7$ out of $7 y^{2}$ .

$y' = \frac{7 (y^{2})}{14 y} + \frac{- 1}{14 y}$

Step 5.5.3.1.1.2

Cancel the common factors.

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Step 5.5.3.1.1.2.1

Factor $7$ out of $14 y$ .

$y' = \frac{7 (y^{2})}{7 (2 y)} + \frac{- 1}{14 y}$

Step 5.5.3.1.1.2.2

Cancel the common factor.

$y' = \frac{7 y^{2}}{7 (2 y)} + \frac{- 1}{14 y}$

Step 5.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{y^{2}}{2 y} + \frac{- 1}{14 y}$

Step 5.5.3.1.2

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

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Step 5.5.3.1.2.1

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

$y' = \frac{y \cdot y}{2 y} + \frac{- 1}{14 y}$

Step 5.5.3.1.2.2

Cancel the common factors.

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Step 5.5.3.1.2.2.1

Factor $y$ out of $2 y$ .

$y' = \frac{y \cdot y}{y \cdot 2} + \frac{- 1}{14 y}$

Step 5.5.3.1.2.2.2

Cancel the common factor.

$y' = \frac{y \cdot y}{y \cdot 2} + \frac{- 1}{14 y}$

Step 5.5.3.1.2.2.3

Rewrite the expression.

$y' = \frac{y}{2} + \frac{- 1}{14 y}$

Step 5.5.3.1.3

Move the negative in front of the fraction.

$y' = \frac{y}{2} - \frac{1}{14 y}$

Step 6

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

$\frac{d y}{d x} = \frac{y}{2} - \frac{1}{14 y}$