Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

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 $g (x) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.3

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

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Step 2.3.1

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

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

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 $f (x) = x^{3}$ and $g (x) = y$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Move $3$ to the left of $y$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $3$ by $2$ .

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

Step 2.4.3

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$1 + y'$

Step 4

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

$x^{2} y' + 2 x y + 2 x^{3} y' + 6 x^{2} y = 1 + y'$

Step 5

Solve for $y'$ .

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Step 5.1

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

$x^{2} y' + 2 x y + 2 x^{3} y' + 6 x^{2} y - y' = 1$

Step 5.2

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

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Step 5.2.1

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

$x^{2} y' + 2 x^{3} y' + 6 x^{2} y - y' = 1 - 2 x y$

Step 5.2.2

Subtract $6 x^{2} y$ from both sides of the equation.

$x^{2} y' + 2 x^{3} y' - y' = 1 - 2 x y - 6 x^{2} y$

Step 5.3

Factor $y'$ out of $x^{2} y' + 2 x^{3} y' - y'$ .

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Step 5.3.1

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

$y' x^{2} + 2 x^{3} y' - y' = 1 - 2 x y - 6 x^{2} y$

Step 5.3.2

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

$y' (x^{2}) + y' (2 x^{3}) - y' = 1 - 2 x y - 6 x^{2} y$

Step 5.3.3

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

$y' (x^{2}) + y' (2 x^{3}) + y' \cdot - 1 = 1 - 2 x y - 6 x^{2} y$

Step 5.3.4

Factor $y'$ out of $y' (x^{2}) + y' (2 x^{3})$ .

$y' (x^{2} + 2 x^{3}) + y' \cdot - 1 = 1 - 2 x y - 6 x^{2} y$

Step 5.3.5

Factor $y'$ out of $y' (x^{2} + 2 x^{3}) + y' \cdot - 1$ .

$y' (x^{2} + 2 x^{3} - 1) = 1 - 2 x y - 6 x^{2} y$

Step 5.4

Reorder terms.

$y' (2 x^{3} + x^{2} - 1) = 1 - 2 x y - 6 x^{2} y$

Step 5.5

Divide each term in $y' (2 x^{3} + x^{2} - 1) = 1 - 2 x y - 6 x^{2} y$ by $2 x^{3} + x^{2} - 1$ and simplify.

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Step 5.5.1

Divide each term in $y' (2 x^{3} + x^{2} - 1) = 1 - 2 x y - 6 x^{2} y$ by $2 x^{3} + x^{2} - 1$ .

$\frac{y' (2 x^{3} + x^{2} - 1)}{2 x^{3} + x^{2} - 1} = \frac{1}{2 x^{3} + x^{2} - 1} + \frac{- 2 x y}{2 x^{3} + x^{2} - 1} + \frac{- 6 x^{2} y}{2 x^{3} + x^{2} - 1}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $2 x^{3} + x^{2} - 1$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{y' (2 x^{3} + x^{2} - 1)}{2 x^{3} + x^{2} - 1} = \frac{1}{2 x^{3} + x^{2} - 1} + \frac{- 2 x y}{2 x^{3} + x^{2} - 1} + \frac{- 6 x^{2} y}{2 x^{3} + x^{2} - 1}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{1}{2 x^{3} + x^{2} - 1} + \frac{- 2 x y}{2 x^{3} + x^{2} - 1} + \frac{- 6 x^{2} y}{2 x^{3} + x^{2} - 1}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Move the negative in front of the fraction.

$y' = \frac{1}{2 x^{3} + x^{2} - 1} - \frac{2 x y}{2 x^{3} + x^{2} - 1} + \frac{- 6 x^{2} y}{2 x^{3} + x^{2} - 1}$

Step 5.5.3.1.2

Move the negative in front of the fraction.

$y' = \frac{1}{2 x^{3} + x^{2} - 1} - \frac{2 x y}{2 x^{3} + x^{2} - 1} - \frac{6 x^{2} y}{2 x^{3} + x^{2} - 1}$

Step 5.5.3.2

Combine into one fraction.

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Step 5.5.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{1 - 2 x y}{2 x^{3} + x^{2} - 1} - \frac{6 x^{2} y}{2 x^{3} + x^{2} - 1}$

Step 5.5.3.2.2

Combine the numerators over the common denominator.

$y' = \frac{1 - 2 x y - 6 x^{2} y}{2 x^{3} + x^{2} - 1}$

Step 6

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

$\frac{d y}{d x} = \frac{1 - 2 x y - 6 x^{2} y}{2 x^{3} + x^{2} - 1}$