Calculus Examples

$y' = \frac{3 y}{x}$ , $y = x^{3}$

Step 1

Find $y'$ .

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Step 1.1

Differentiate both sides of the equation.

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

Step 1.2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 1.3

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

$3 x^{2}$

Step 1.4

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

$y' = 3 x^{2}$

Step 2

Substitute into the given differential equation.

$3 x^{2} = \frac{3 x^{3}}{x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Raise $x$ to the power of $1$ .

$3 x^{2} = \frac{x (3 x^{2})}{x^{1}}$

Step 3.2.2

Factor $x$ out of $x^{1}$ .

$3 x^{2} = \frac{x (3 x^{2})}{x \cdot 1}$

Step 3.2.3

Cancel the common factor.

$3 x^{2} = \frac{x (3 x^{2})}{x \cdot 1}$

Step 3.2.4

Rewrite the expression.

$3 x^{2} = \frac{3 x^{2}}{1}$

Step 3.2.5

Divide $3 x^{2}$ by $1$ .

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

Step 4

The given solution satisfies the given differential equation.

$y = x^{3}$ is a solution to $y' = \frac{3 y}{x}$