Calculus Examples

$\frac{X - 1}{X}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d X} [\frac{f (X)}{g (X)}]$ is $\frac{g (X) \frac{d}{d X} [f (X)] - f (X) \frac{d}{d X} [g (X)]}{{g (X)}^{2}}$ where $f (X) = X - 1$ and $g (X) = X$ .

$\frac{X \frac{d}{d X} [X - 1] - (X - 1) \frac{d}{d X} [X]}{X^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $X - 1$ with respect to $X$ is $\frac{d}{d X} [X] + \frac{d}{d X} [- 1]$ .

$\frac{X (\frac{d}{d X} [X] + \frac{d}{d X} [- 1]) - (X - 1) \frac{d}{d X} [X]}{X^{2}}$

Step 2.2

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

$\frac{X (1 + \frac{d}{d X} [- 1]) - (X - 1) \frac{d}{d X} [X]}{X^{2}}$

Step 2.3

Since $- 1$ is constant with respect to $X$ , the derivative of $- 1$ with respect to $X$ is $0$ .

$\frac{X (1 + 0) - (X - 1) \frac{d}{d X} [X]}{X^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

$\frac{X \cdot 1 - (X - 1) \frac{d}{d X} [X]}{X^{2}}$

Step 2.4.2

Multiply $X$ by $1$ .

$\frac{X - (X - 1) \frac{d}{d X} [X]}{X^{2}}$

Step 2.5

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

$\frac{X - (X - 1) \cdot 1}{X^{2}}$

Step 2.6

Multiply $- 1$ by $1$ .

$\frac{X - (X - 1)}{X^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{X - X - - 1}{X^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Subtract $X$ from $X$ .

$\frac{0 - - 1}{X^{2}}$

Step 3.2.2

Subtract $- 1$ from $0$ .

$\frac{- - 1}{X^{2}}$

Step 3.2.3

Multiply $- 1$ by $- 1$ .

$\frac{1}{X^{2}}$