Calculus Examples

$x^{2} y + y x^{2} = 6$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x^{2}$ and $g (y) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.3

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

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Step 2.3.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y$ and $g (y) = x^{2}$ .

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

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

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

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

Step 2.4

Simplify.

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Step 2.4.1

Combine terms.

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Step 2.4.1.1

Add $2 y x x'$ and $2 y x x'$ .

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

Step 2.4.1.2

Add $x^{2}$ and $x^{2}$ .

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

Step 2.4.2

Reorder terms.

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

Step 3

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

$0$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

Cancel the common factor of $x$ .

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Rewrite the expression.

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

Step 5.2.2.3

Cancel the common factor of $y$ .

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Step 5.2.2.3.1

Cancel the common factor.

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

Step 5.2.2.3.2

Divide $x'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

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

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Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

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

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

Step 5.2.3.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.3

Rewrite the expression.

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

Step 5.2.3.2

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

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Step 5.2.3.2.1

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

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

Step 5.2.3.2.2

Cancel the common factors.

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Step 5.2.3.2.2.1

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

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

Step 5.2.3.2.2.2

Cancel the common factor.

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

Step 5.2.3.2.2.3

Rewrite the expression.

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

Step 5.2.3.3

Move the negative in front of the fraction.

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

Step 6

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

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