Calculus Examples

$y = \frac{x^{5}}{x^{3}}$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Cancel the common factor of $x^{5}$ and $x^{3}$ .

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Step 3.1.1

Factor $x^{3}$ out of $x^{5}$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Multiply by $1$ .

$\frac{d}{d x} [\frac{x^{3} x^{2}}{x^{3} \cdot 1}]$

Step 3.1.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{x^{3} x^{2}}{x^{3} \cdot 1}]$

Step 3.1.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{x^{2}}{1}]$

Step 3.1.2.4

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

$\frac{d}{d x} [x^{2}]$

Step 3.2

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

$2 x$

Step 4

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

$y' = 2 x$

Step 5

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

$\frac{d y}{d x} = 2 x$