Calculus Examples

$y^{2} + 2 x y = x^{3} - x^{2} + 17$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2} + 2 x y) = \frac{d}{d x} (x^{3} - x^{2} + 17)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 2.3

Evaluate $\frac{d}{d x} [2 x y]$ .

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Step 2.3.1

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

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

Step 2.3.2

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 $g (x) = y$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

$2 y y' + 2 (x y' + y \cdot 1)$

Step 2.3.5

Multiply $y$ by $1$ .

$2 y y' + 2 (x y' + y)$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$2 y y' + 2 (x y') + 2 y$

Step 2.4.2

Remove unnecessary parentheses.

$2 y y' + 2 x y' + 2 y$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $x^{3} - x^{2} + 17$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [17]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [17]$

Step 3.1.2

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

$3 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [17]$

Step 3.2

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [17]$

Step 3.2.2

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

$3 x^{2} - (2 x) + \frac{d}{d x} [17]$

Step 3.2.3

Multiply $2$ by $- 1$ .

$3 x^{2} - 2 x + \frac{d}{d x} [17]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $17$ is constant with respect to $x$ , the derivative of $17$ with respect to $x$ is $0$ .

$3 x^{2} - 2 x + 0$

Step 3.3.2

Add $3 x^{2} - 2 x$ and $0$ .

$3 x^{2} - 2 x$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.3

Divide each term in $2 y' (y + x) = 3 x^{2} - 2 x - 2 y$ by $2 (y + x)$ and simplify.

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Step 5.3.1

Divide each term in $2 y' (y + x) = 3 x^{2} - 2 x - 2 y$ by $2 (y + x)$ .

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $y + x$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $- 2$ and $2$ .

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Step 5.3.3.1.1.1

Factor $2$ out of $- 2 x$ .

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

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Cancel the common factor.

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

Step 5.3.3.1.1.2.2

Rewrite the expression.

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

Step 5.3.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.3.1.3

Cancel the common factor of $- 2$ and $2$ .

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Step 5.3.3.1.3.1

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

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

Step 5.3.3.1.3.2

Cancel the common factors.

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Step 5.3.3.1.3.2.1

Cancel the common factor.

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

Step 5.3.3.1.3.2.2

Rewrite the expression.

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

Step 5.3.3.1.4

Move the negative in front of the fraction.

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

Step 5.3.3.2

To write $- \frac{x}{y + x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.3.3.3

Write each expression with a common denominator of $2 (y + x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.3.1

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

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

Step 5.3.3.3.2

Reorder the factors of $(y + x) \cdot 2$ .

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

Step 5.3.3.4

Combine the numerators over the common denominator.

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

Step 5.3.3.5

Simplify the numerator.

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Step 5.3.3.5.1

Factor $x$ out of $3 x^{2} - x \cdot 2$ .

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Step 5.3.3.5.1.1

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

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

Step 5.3.3.5.1.2

Factor $x$ out of $- x \cdot 2$ .

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

Step 5.3.3.5.1.3

Factor $x$ out of $x (3 x) + x (- 1 \cdot 2)$ .

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

Step 5.3.3.5.2

Multiply $- 1$ by $2$ .

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

Step 5.3.3.6

To write $- \frac{y}{y + x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.3.3.7

Write each expression with a common denominator of $2 (y + x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.7.1

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

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

Step 5.3.3.7.2

Reorder the factors of $(y + x) \cdot 2$ .

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

Step 5.3.3.8

Combine the numerators over the common denominator.

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

Step 5.3.3.9

Simplify the numerator.

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Step 5.3.3.9.1

Apply the distributive property.

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

Step 5.3.3.9.2

Rewrite using the commutative property of multiplication.

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

Step 5.3.3.9.3

Move $- 2$ to the left of $x$ .

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

Step 5.3.3.9.4

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.3.9.4.1

Move $x$ .

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

Step 5.3.3.9.4.2

Multiply $x$ by $x$ .

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

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

Step 5.3.3.9.5

Multiply $2$ by $- 1$ .

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

Step 6

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

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