Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$1 + y' + 0$

Step 2.5

Add $1 + y'$ and $0$ .

$1 + y'$

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

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 $y$ .

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

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 $n = 2$ .

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify.

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Step 3.5.1

Reorder the factors of $\frac{1}{x^{2} + y^{2}} (2 x + 2 y y')$ .

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

Step 3.5.2

Multiply $2 x + 2 y y'$ by $\frac{1}{x^{2} + y^{2}}$ .

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

Step 3.5.3

Factor $2$ out of $2 x + 2 y y'$ .

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Step 3.5.3.1

Factor $2$ out of $2 x$ .

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

Step 3.5.3.2

Factor $2$ out of $2 y y'$ .

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

Step 3.5.3.3

Factor $2$ out of $2 (x) + 2 (y y')$ .

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $x^{2} + y^{2}$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $(1 + y') (x^{2} + y^{2})$ .

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Step 5.2.1.1.1

Expand $(1 + y') (x^{2} + y^{2})$ using the FOIL Method.

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Step 5.2.1.1.1.1

Apply the distributive property.

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

Step 5.2.1.1.1.2

Apply the distributive property.

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

Step 5.2.1.1.1.3

Apply the distributive property.

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

Step 5.2.1.1.2

Simplify terms.

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Step 5.2.1.1.2.1

Simplify each term.

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Step 5.2.1.1.2.1.1

Multiply $x^{2}$ by $1$ .

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

Step 5.2.1.1.2.1.2

Multiply $y^{2}$ by $1$ .

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

Step 5.2.1.1.2.2

Reorder.

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Step 5.2.1.1.2.2.1

Move $y^{2}$ .

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

Step 5.2.1.1.2.2.2

Move $x^{2}$ .

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

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

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Step 5.2.2.1.1.1

Cancel the common factor.

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

Step 5.2.2.1.1.2

Rewrite the expression.

$y' x^{2} + y' y^{2} + x^{2} + y^{2} = 2 (x + y y')$

Step 5.2.2.1.2

Apply the distributive property.

$y' x^{2} + y' y^{2} + x^{2} + y^{2} = 2 x + 2 (y y')$

Step 5.2.2.1.3

Simplify the expression.

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Step 5.2.2.1.3.1

Move $y$ .

$y' x^{2} + y' y^{2} + x^{2} + y^{2} = 2 x + 2 y' y$

Step 5.2.2.1.3.2

Reorder $2 x$ and $2 y' y$ .

$y' x^{2} + y' y^{2} + x^{2} + y^{2} = 2 y' y + 2 x$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Subtract $2 y' y$ from both sides of the equation.

$y' x^{2} + y' y^{2} + x^{2} + y^{2} - 2 y' y = 2 x$

Step 5.3.2

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

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Step 5.3.2.1

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

$y' x^{2} + y' y^{2} + y^{2} - 2 y' y = 2 x - x^{2}$

Step 5.3.2.2

Subtract $y^{2}$ from both sides of the equation.

$y' x^{2} + y' y^{2} - 2 y' y = 2 x - x^{2} - y^{2}$

Step 5.3.3

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

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Step 5.3.3.1

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

$y' (x^{2}) + y' (y^{2}) - 2 y' y = 2 x - x^{2} - y^{2}$

Step 5.3.3.2

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

$y' (x^{2}) + y' (y^{2}) - 2 y' y = 2 x - x^{2} - y^{2}$

Step 5.3.3.3

Factor $y'$ out of $- 2 y' y$ .

$y' (x^{2}) + y' (y^{2}) + y' (- 2 y) = 2 x - x^{2} - y^{2}$

Step 5.3.3.4

Factor $y'$ out of $y' (x^{2}) + y' (y^{2})$ .

$y' (x^{2} + y^{2}) + y' (- 2 y) = 2 x - x^{2} - y^{2}$

Step 5.3.3.5

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

$y' (x^{2} + y^{2} - 2 y) = 2 x - x^{2} - y^{2}$

Step 5.3.4

Divide each term in $y' (x^{2} + y^{2} - 2 y) = 2 x - x^{2} - y^{2}$ by $x^{2} + y^{2} - 2 y$ and simplify.

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Step 5.3.4.1

Divide each term in $y' (x^{2} + y^{2} - 2 y) = 2 x - x^{2} - y^{2}$ by $x^{2} + y^{2} - 2 y$ .

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

Step 5.3.4.2

Simplify the left side.

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Step 5.3.4.2.1

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

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Step 5.3.4.2.1.1

Cancel the common factor.

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

Step 5.3.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.4.3

Simplify the right side.

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Step 5.3.4.3.1

Simplify each term.

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Step 5.3.4.3.1.1

Move the negative in front of the fraction.

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

Step 5.3.4.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.4.3.2

Combine into one fraction.

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Step 5.3.4.3.2.1

Combine the numerators over the common denominator.

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

Step 5.3.4.3.2.2

Combine the numerators over the common denominator.

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

Step 6

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

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