Calculus Examples

$x y^{4} + x^{2} y = x + 3 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x y^{4} + x^{2} y) = \frac{d}{d x} (x + 3 y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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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 $g (x) = y^{4}$ .

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

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 $f (x) = x^{4}$ and $g (x) = y$ .

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Step 2.2.2.1

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

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

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 $n = 4$ .

$x (4 u^{3} \frac{d}{d x} [y]) + y^{4} \frac{d}{d x} [x] + \frac{d}{d x} [x^{2} y]$

Step 2.2.2.3

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

$x (4 y^{3} \frac{d}{d x} [y]) + y^{4} \frac{d}{d x} [x] + \frac{d}{d x} [x^{2} y]$

Step 2.2.3

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

$x (4 y^{3} y') + y^{4} \frac{d}{d x} [x] + \frac{d}{d x} [x^{2} y]$

Step 2.2.4

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

$x (4 y^{3} y') + y^{4} \cdot 1 + \frac{d}{d x} [x^{2} y]$

Step 2.2.5

Move $4$ to the left of $x$ .

$4 \cdot x y^{3} y' + y^{4} \cdot 1 + \frac{d}{d x} [x^{2} y]$

Step 2.2.6

Multiply $y^{4}$ by $1$ .

$4 x y^{3} y' + y^{4} + \frac{d}{d x} [x^{2} y]$

Step 2.3

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

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Step 2.3.1

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 $g (x) = y$ .

$4 x y^{3} y' + y^{4} + x^{2} \frac{d}{d x} [y] + y \frac{d}{d x} [x^{2}]$

Step 2.3.2

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

$4 x y^{3} y' + y^{4} + x^{2} y' + y \frac{d}{d x} [x^{2}]$

Step 2.3.3

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

$4 x y^{3} y' + y^{4} + x^{2} y' + y (2 x)$

Step 2.3.4

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

$4 x y^{3} y' + y^{4} + x^{2} y' + 2 y x$

Step 2.4

Reorder terms.

$y^{4} + 4 y^{3} x y' + x^{2} y' + 2 x y$

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 $x + 3 y$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3 y]$ .

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

Step 3.1.2

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

$1 + 3 y'$

Step 3.3

Reorder terms.

$3 y' + 1$

Step 4

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

$y^{4} + 4 y^{3} x y' + x^{2} y' + 2 x y = 3 y' + 1$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $3 y'$ from both sides of the equation.

$y^{4} + 4 y^{3} x y' + x^{2} y' + 2 x y - 3 y' = 1$

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 $y^{4}$ from both sides of the equation.

$4 y^{3} x y' + x^{2} y' + 2 x y - 3 y' = 1 - y^{4}$

Step 5.2.2

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

$4 y^{3} x y' + x^{2} y' - 3 y' = 1 - y^{4} - 2 x y$

Step 5.3

Factor $y'$ out of $4 y^{3} x y' + x^{2} y' - 3 y'$ .

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Step 5.3.1

Factor $y'$ out of $4 y^{3} x y'$ .

$y' (4 y^{3} x) + x^{2} y' - 3 y' = 1 - y^{4} - 2 x y$

Step 5.3.2

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

$y' (4 y^{3} x) + y' (x^{2}) - 3 y' = 1 - y^{4} - 2 x y$

Step 5.3.3

Factor $y'$ out of $- 3 y'$ .

$y' (4 y^{3} x) + y' (x^{2}) + y' \cdot - 3 = 1 - y^{4} - 2 x y$

Step 5.3.4

Factor $y'$ out of $y' (4 y^{3} x) + y' (x^{2})$ .

$y' (4 y^{3} x + x^{2}) + y' \cdot - 3 = 1 - y^{4} - 2 x y$

Step 5.3.5

Factor $y'$ out of $y' (4 y^{3} x + x^{2}) + y' \cdot - 3$ .

$y' (4 y^{3} x + x^{2} - 3) = 1 - y^{4} - 2 x y$

Step 5.4

Divide each term in $y' (4 y^{3} x + x^{2} - 3) = 1 - y^{4} - 2 x y$ by $4 y^{3} x + x^{2} - 3$ and simplify.

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Step 5.4.1

Divide each term in $y' (4 y^{3} x + x^{2} - 3) = 1 - y^{4} - 2 x y$ by $4 y^{3} x + x^{2} - 3$ .

$\frac{y' (4 y^{3} x + x^{2} - 3)}{4 y^{3} x + x^{2} - 3} = \frac{1}{4 y^{3} x + x^{2} - 3} + \frac{- y^{4}}{4 y^{3} x + x^{2} - 3} + \frac{- 2 x y}{4 y^{3} x + x^{2} - 3}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $4 y^{3} x + x^{2} - 3$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (4 y^{3} x + x^{2} - 3)}{4 y^{3} x + x^{2} - 3} = \frac{1}{4 y^{3} x + x^{2} - 3} + \frac{- y^{4}}{4 y^{3} x + x^{2} - 3} + \frac{- 2 x y}{4 y^{3} x + x^{2} - 3}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{1}{4 y^{3} x + x^{2} - 3} + \frac{- y^{4}}{4 y^{3} x + x^{2} - 3} + \frac{- 2 x y}{4 y^{3} x + x^{2} - 3}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Move the negative in front of the fraction.

$y' = \frac{1}{4 y^{3} x + x^{2} - 3} - \frac{y^{4}}{4 y^{3} x + x^{2} - 3} + \frac{- 2 x y}{4 y^{3} x + x^{2} - 3}$

Step 5.4.3.1.2

Move the negative in front of the fraction.

$y' = \frac{1}{4 y^{3} x + x^{2} - 3} - \frac{y^{4}}{4 y^{3} x + x^{2} - 3} - \frac{2 x y}{4 y^{3} x + x^{2} - 3}$

Step 5.4.3.2

Combine into one fraction.

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Step 5.4.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{1 - y^{4}}{4 y^{3} x + x^{2} - 3} - \frac{2 x y}{4 y^{3} x + x^{2} - 3}$

Step 5.4.3.2.2

Combine the numerators over the common denominator.

$y' = \frac{1 - y^{4} - 2 x y}{4 y^{3} x + x^{2} - 3}$

Step 6

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

$\frac{d y}{d x} = \frac{1 - y^{4} - 2 x y}{4 y^{3} x + x^{2} - 3}$