Calculus Examples

$y + x^{3} y - 5 x = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y + x^{3} y - 5 x) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $y + x^{3} y - 5 x$ with respect to $x$ is $\frac{d}{d x} [y] + \frac{d}{d x} [x^{3} y] + \frac{d}{d x} [- 5 x]$ .

$\frac{d}{d x} [y] + \frac{d}{d x} [x^{3} y] + \frac{d}{d x} [- 5 x]$

Step 2.2

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

$y' + \frac{d}{d x} [x^{3} y] + \frac{d}{d x} [- 5 x]$

Step 2.3

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

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

$y' + x^{3} \frac{d}{d x} [y] + y \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 5 x]$

Step 2.3.2

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

$y' + x^{3} y' + y \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 5 x]$

Step 2.3.3

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

$y' + x^{3} y' + y (3 x^{2}) + \frac{d}{d x} [- 5 x]$

Step 2.3.4

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

$y' + x^{3} y' + 3 y x^{2} + \frac{d}{d x} [- 5 x]$

Step 2.4

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 2.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$y' + x^{3} y' + 3 y x^{2} - 5 \frac{d}{d x} [x]$

Step 2.4.2

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

$y' + x^{3} y' + 3 y x^{2} - 5 \cdot 1$

Step 2.4.3

Multiply $- 5$ by $1$ .

$y' + x^{3} y' + 3 y x^{2} - 5$

Step 2.5

Reorder terms.

$x^{3} y' + 3 x^{2} y + y' - 5$

Step 3

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

$0$

Step 4

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

$x^{3} y' + 3 x^{2} y + y' - 5 = 0$

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 $3 x^{2} y$ from both sides of the equation.

$x^{3} y' + y' - 5 = - 3 x^{2} y$

Step 5.1.2

Add $5$ to both sides of the equation.

$x^{3} y' + y' = - 3 x^{2} y + 5$

Step 5.2

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

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Step 5.2.1

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

$y' x^{3} + y' = - 3 x^{2} y + 5$

Step 5.2.2

Raise $y'$ to the power of $1$ .

$y' x^{3} + y' = - 3 x^{2} y + 5$

Step 5.2.3

Factor $y'$ out of ${y'}^{1}$ .

$y' x^{3} + y' \cdot 1 = - 3 x^{2} y + 5$

Step 5.2.4

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

$y' (x^{3} + 1) = - 3 x^{2} y + 5$

Step 5.3

Rewrite $1$ as $1^{3}$ .

$y' (x^{3} + 1^{3}) = - 3 x^{2} y + 5$

Step 5.4

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

$y' ((x + 1) (x^{2} - x \cdot 1 + 1^{2})) = - 3 x^{2} y + 5$

Step 5.5

Factor.

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Step 5.5.1

Simplify.

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Step 5.5.1.1

Multiply $- 1$ by $1$ .

$y' ((x + 1) (x^{2} - x + 1^{2})) = - 3 x^{2} y + 5$

Step 5.5.1.2

One to any power is one.

$y' ((x + 1) (x^{2} - x + 1)) = - 3 x^{2} y + 5$

Step 5.5.2

Remove unnecessary parentheses.

$y' (x + 1) (x^{2} - x + 1) = - 3 x^{2} y + 5$

Step 5.6

Divide each term in $y' (x + 1) (x^{2} - x + 1) = - 3 x^{2} y + 5$ by $(x + 1) (x^{2} - x + 1)$ and simplify.

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Step 5.6.1

Divide each term in $y' (x + 1) (x^{2} - x + 1) = - 3 x^{2} y + 5$ by $(x + 1) (x^{2} - x + 1)$ .

$\frac{y' (x + 1) (x^{2} - x + 1)}{(x + 1) (x^{2} - x + 1)} = \frac{- 3 x^{2} y}{(x + 1) (x^{2} - x + 1)} + \frac{5}{(x + 1) (x^{2} - x + 1)}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $x + 1$ .

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Step 5.6.2.1.1

Cancel the common factor.

$\frac{y' (x + 1) (x^{2} - x + 1)}{(x + 1) (x^{2} - x + 1)} = \frac{- 3 x^{2} y}{(x + 1) (x^{2} - x + 1)} + \frac{5}{(x + 1) (x^{2} - x + 1)}$

Step 5.6.2.1.2

Rewrite the expression.

$\frac{y' (x^{2} - x + 1)}{x^{2} - x + 1} = \frac{- 3 x^{2} y}{(x + 1) (x^{2} - x + 1)} + \frac{5}{(x + 1) (x^{2} - x + 1)}$

Step 5.6.2.2

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

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Step 5.6.2.2.1

Cancel the common factor.

$\frac{y' (x^{2} - x + 1)}{x^{2} - x + 1} = \frac{- 3 x^{2} y}{(x + 1) (x^{2} - x + 1)} + \frac{5}{(x + 1) (x^{2} - x + 1)}$

Step 5.6.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 3 x^{2} y}{(x + 1) (x^{2} - x + 1)} + \frac{5}{(x + 1) (x^{2} - x + 1)}$

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Move the negative in front of the fraction.

$y' = - \frac{3 x^{2} y}{(x + 1) (x^{2} - x + 1)} + \frac{5}{(x + 1) (x^{2} - x + 1)}$

Step 6

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

$\frac{d y}{d x} = - \frac{3 x^{2} y}{(x + 1) (x^{2} - x + 1)} + \frac{5}{(x + 1) (x^{2} - x + 1)}$