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

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

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