Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Reorder.

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Step 2.4.1

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

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

Step 2.4.2

Reorder terms.

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

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 $- 5 - 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- 5] + \frac{d}{d x} [- 3 x^{2}]$ .

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

Step 3.1.2

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 3 x^{2}]$

Step 3.2

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

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Step 3.2.1

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

$0 - 3 \frac{d}{d x} [x^{2}]$

Step 3.2.2

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

$0 - 3 (2 x)$

Step 3.2.3

Multiply $2$ by $- 3$ .

$0 - 6 x$

Step 3.3

Subtract $6 x$ from $0$ .

$- 6 x$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $x^{3}$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $y'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Cancel the common factor of $x$ and $x^{3}$ .

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Step 5.2.3.1.1.1

Factor $x$ out of $- 6 x$ .

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

Step 5.2.3.1.1.2

Cancel the common factors.

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Step 5.2.3.1.1.2.1

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

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

Step 5.2.3.1.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.1.2.3

Rewrite the expression.

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

Step 5.2.3.1.2

Move the negative in front of the fraction.

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

Step 5.2.3.1.3

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 5.2.3.1.3.1

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

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

Step 5.2.3.1.3.2

Cancel the common factors.

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Step 5.2.3.1.3.2.1

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

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

Step 5.2.3.1.3.2.2

Cancel the common factor.

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

Step 5.2.3.1.3.2.3

Rewrite the expression.

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

Step 5.2.3.1.4

Move the negative in front of the fraction.

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

Step 6

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

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