Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

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

Step 2.2.2

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

$x^{2} y' + y \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 y^{3}]$

Step 2.2.3

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

$x^{2} y' + y (2 x) + \frac{d}{d x} [2 y^{3}]$

Step 2.2.4

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

$x^{2} y' + 2 y x + \frac{d}{d x} [2 y^{3}]$

Step 2.3

Evaluate $\frac{d}{d x} [2 y^{3}]$ .

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Step 2.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 y^{3}$ with respect to $x$ is $2 \frac{d}{d x} [y^{3}]$ .

$x^{2} y' + 2 y x + 2 \frac{d}{d x} [y^{3}]$

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.3.2.1

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

$x^{2} y' + 2 y x + 2 (\frac{d}{d u} [u^{3}] \frac{d}{d x} [y])$

Step 2.3.2.2

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

$x^{2} y' + 2 y x + 2 (3 u^{2} \frac{d}{d x} [y])$

Step 2.3.2.3

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

$x^{2} y' + 2 y x + 2 (3 y^{2} \frac{d}{d x} [y])$

Step 2.3.3

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

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

Step 2.3.4

Multiply $3$ by $2$ .

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

Step 2.4

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$\frac{d}{d x} [3 x] + \frac{d}{d x} [2 y]$

Step 3.2

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 3.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [2 y]$

Step 3.2.2

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

$3 \cdot 1 + \frac{d}{d x} [2 y]$

Step 3.2.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [2 y]$

Step 3.3

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

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Step 3.3.1

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

$3 + 2 \frac{d}{d x} [y]$

Step 3.3.2

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

$3 + 2 y'$

Step 3.4

Reorder terms.

$2 y' + 3$

Step 4

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

$x^{2} y' + 2 x y + 6 y^{2} y' = 2 y' + 3$

Step 5

Solve for $y'$ .

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Step 5.1

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

$x^{2} y' + 2 x y + 6 y^{2} y' - 2 y' = 3$

Step 5.2

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

$x^{2} y' + 6 y^{2} y' - 2 y' = 3 - 2 x y$

Step 5.3

Factor $y'$ out of $x^{2} y' + 6 y^{2} y' - 2 y'$ .

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Step 5.3.1

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

$y' x^{2} + 6 y^{2} y' - 2 y' = 3 - 2 x y$

Step 5.3.2

Factor $y'$ out of $6 y^{2} y'$ .

$y' (x^{2}) + y' (6 y^{2}) - 2 y' = 3 - 2 x y$

Step 5.3.3

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

$y' (x^{2}) + y' (6 y^{2}) + y' \cdot - 2 = 3 - 2 x y$

Step 5.3.4

Factor $y'$ out of $y' (x^{2}) + y' (6 y^{2})$ .

$y' (x^{2} + 6 y^{2}) + y' \cdot - 2 = 3 - 2 x y$

Step 5.3.5

Factor $y'$ out of $y' (x^{2} + 6 y^{2}) + y' \cdot - 2$ .

$y' (x^{2} + 6 y^{2} - 2) = 3 - 2 x y$

Step 5.4

Divide each term in $y' (x^{2} + 6 y^{2} - 2) = 3 - 2 x y$ by $x^{2} + 6 y^{2} - 2$ and simplify.

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Step 5.4.1

Divide each term in $y' (x^{2} + 6 y^{2} - 2) = 3 - 2 x y$ by $x^{2} + 6 y^{2} - 2$ .

$\frac{y' (x^{2} + 6 y^{2} - 2)}{x^{2} + 6 y^{2} - 2} = \frac{3}{x^{2} + 6 y^{2} - 2} + \frac{- 2 x y}{x^{2} + 6 y^{2} - 2}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x^{2} + 6 y^{2} - 2$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (x^{2} + 6 y^{2} - 2)}{x^{2} + 6 y^{2} - 2} = \frac{3}{x^{2} + 6 y^{2} - 2} + \frac{- 2 x y}{x^{2} + 6 y^{2} - 2}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{3}{x^{2} + 6 y^{2} - 2} + \frac{- 2 x y}{x^{2} + 6 y^{2} - 2}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$y' = \frac{3 - 2 x y}{x^{2} + 6 y^{2} - 2}$

Step 6

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

$\frac{d y}{d x} = \frac{3 - 2 x y}{x^{2} + 6 y^{2} - 2}$