Calculus Examples

$x y^{2} + 4 x y = 10$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x y^{2} + 4 x y) = \frac{d}{d x} (10)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $x y^{2} + 4 x y$ with respect to $x$ is $\frac{d}{d x} [x y^{2}] + \frac{d}{d x} [4 x y]$ .

$\frac{d}{d x} [x y^{2}] + \frac{d}{d x} [4 x y]$

Step 2.2

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

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Step 2.2.1

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

$x \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [4 x y]$

Step 2.2.2

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

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Step 2.2.2.1

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

$x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [4 x y]$

Step 2.2.2.2

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

$x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [4 x y]$

Step 2.2.2.3

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

$x (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [4 x y]$

Step 2.2.3

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

$x (2 y y') + y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [4 x y]$

Step 2.2.4

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

$x (2 y y') + y^{2} \cdot 1 + \frac{d}{d x} [4 x y]$

Step 2.2.5

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

$2 \cdot x y y' + y^{2} \cdot 1 + \frac{d}{d x} [4 x y]$

Step 2.2.6

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

$2 x y y' + y^{2} + \frac{d}{d x} [4 x y]$

Step 2.3

Evaluate $\frac{d}{d x} [4 x y]$ .

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Step 2.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x y$ with respect to $x$ is $4 \frac{d}{d x} [x y]$ .

$2 x y y' + y^{2} + 4 \frac{d}{d x} [x y]$

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 $g (x) = y$ .

$2 x y y' + y^{2} + 4 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 2.3.3

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

$2 x y y' + y^{2} + 4 (x y' + y \frac{d}{d x} [x])$

Step 2.3.4

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

$2 x y y' + y^{2} + 4 (x y' + y \cdot 1)$

Step 2.3.5

Multiply $y$ by $1$ .

$2 x y y' + y^{2} + 4 (x y' + y)$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$2 x y y' + y^{2} + 4 (x y') + 4 y$

Step 2.4.2

Remove unnecessary parentheses.

$2 x y y' + y^{2} + 4 x y' + 4 y$

Step 2.4.3

Reorder terms.

$y^{2} + 2 x y y' + 4 x y' + 4 y$

Step 3

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$0$

Step 4

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

$y^{2} + 2 x y y' + 4 x y' + 4 y = 0$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

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

$2 x y y' + 4 x y' + 4 y = - y^{2}$

Step 5.1.2

Subtract $4 y$ from both sides of the equation.

$2 x y y' + 4 x y' = - y^{2} - 4 y$

Step 5.2

Factor $2 x y'$ out of $2 x y y' + 4 x y'$ .

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Step 5.2.1

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

$2 x y' y + 4 x y' = - y^{2} - 4 y$

Step 5.2.2

Factor $2 x y'$ out of $4 x y'$ .

$2 x y' y + 2 x y' \cdot 2 = - y^{2} - 4 y$

Step 5.2.3

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

$2 x y' (y + 2) = - y^{2} - 4 y$

Step 5.3

Divide each term in $2 x y' (y + 2) = - y^{2} - 4 y$ by $2 x (y + 2)$ and simplify.

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Step 5.3.1

Divide each term in $2 x y' (y + 2) = - y^{2} - 4 y$ by $2 x (y + 2)$ .

$\frac{2 x y' (y + 2)}{2 x (y + 2)} = \frac{- y^{2}}{2 x (y + 2)} + \frac{- 4 y}{2 x (y + 2)}$

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.

$\frac{2 x y' (y + 2)}{2 x (y + 2)} = \frac{- y^{2}}{2 x (y + 2)} + \frac{- 4 y}{2 x (y + 2)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{x y' (y + 2)}{x (y + 2)} = \frac{- y^{2}}{2 x (y + 2)} + \frac{- 4 y}{2 x (y + 2)}$

Step 5.3.2.2

Cancel the common factor of $x$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{x y' (y + 2)}{x (y + 2)} = \frac{- y^{2}}{2 x (y + 2)} + \frac{- 4 y}{2 x (y + 2)}$

Step 5.3.2.2.2

Rewrite the expression.

$\frac{y' (y + 2)}{y + 2} = \frac{- y^{2}}{2 x (y + 2)} + \frac{- 4 y}{2 x (y + 2)}$

Step 5.3.2.3

Cancel the common factor of $y + 2$ .

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Step 5.3.2.3.1

Cancel the common factor.

$\frac{y' (y + 2)}{y + 2} = \frac{- y^{2}}{2 x (y + 2)} + \frac{- 4 y}{2 x (y + 2)}$

Step 5.3.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{- y^{2}}{2 x (y + 2)} + \frac{- 4 y}{2 x (y + 2)}$

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

Move the negative in front of the fraction.

$y' = - \frac{y^{2}}{2 x (y + 2)} + \frac{- 4 y}{2 x (y + 2)}$

Step 5.3.3.1.2

Cancel the common factor of $- 4$ and $2$ .

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Step 5.3.3.1.2.1

Factor $2$ out of $- 4 y$ .

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

Step 5.3.3.1.2.2

Cancel the common factors.

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Step 5.3.3.1.2.2.1

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

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

Step 5.3.3.1.2.2.2

Cancel the common factor.

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

Step 5.3.3.1.2.2.3

Rewrite the expression.

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

Step 5.3.3.1.3

Move the negative in front of the fraction.

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

Step 6

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

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