Calculus Examples

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

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 $f (x) = 2 x - 1$ and $g (x) = x^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [2 x - 1] - (2 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$\frac{x^{2} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]) - (2 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3

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

$\frac{x^{2} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (2 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.4

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

$\frac{x^{2} (2 \cdot 1 + \frac{d}{d x} [- 1]) - (2 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5

Multiply $2$ by $1$ .

$\frac{x^{2} (2 + \frac{d}{d x} [- 1]) - (2 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6

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

$\frac{x^{2} (2 + 0) - (2 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.7

Simplify the expression.

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Step 2.7.1

Add $2$ and $0$ .

$\frac{x^{2} \cdot 2 - (2 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.7.2

Move $2$ to the left of $x^{2}$ .

$\frac{2 \cdot x^{2} - (2 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.8

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

$\frac{2 x^{2} - (2 x - 1) (2 x)}{x^{4}}$

Step 2.9

Multiply $2$ by $- 1$ .

$\frac{2 x^{2} - 2 (2 x - 1) x}{x^{4}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{2 x^{2} + (- 2 (2 x) - 2 \cdot - 1) x}{x^{4}}$

Step 3.2

Apply the distributive property.

$\frac{2 x^{2} - 2 (2 x) x - 2 \cdot - 1 x}{x^{4}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.1.1

Move $x$ .

$\frac{2 x^{2} - 2 (2 (x \cdot x)) - 2 \cdot - 1 x}{x^{4}}$

Step 3.3.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 2 (2 x^{2}) - 2 \cdot - 1 x}{x^{4}}$

Step 3.3.1.2

Multiply $2$ by $- 2$ .

$\frac{2 x^{2} - 4 x^{2} - 2 \cdot - 1 x}{x^{4}}$

Step 3.3.1.3

Multiply $- 2$ by $- 1$ .

$\frac{2 x^{2} - 4 x^{2} + 2 x}{x^{4}}$

Step 3.3.2

Subtract $4 x^{2}$ from $2 x^{2}$ .

$\frac{- 2 x^{2} + 2 x}{x^{4}}$

Step 3.4

Factor $2 x$ out of $- 2 x^{2} + 2 x$ .

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Step 3.4.1

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

$\frac{2 x (- x) + 2 x}{x^{4}}$

Step 3.4.2

Factor $2 x$ out of $2 x$ .

$\frac{2 x (- x) + 2 x (1)}{x^{4}}$

Step 3.4.3

Factor $2 x$ out of $2 x (- x) + 2 x (1)$ .

$\frac{2 x (- x + 1)}{x^{4}}$

Step 3.5

Cancel the common factor of $x$ and $x^{4}$ .

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Step 3.5.1

Factor $x$ out of $2 x (- x + 1)$ .

$\frac{x (2 (- x + 1))}{x^{4}}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $x$ out of $x^{4}$ .

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

Step 3.5.2.2

Cancel the common factor.

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

Step 3.5.2.3

Rewrite the expression.

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

Step 3.6

Factor $- 1$ out of $- x$ .

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

Step 3.7

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.8

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

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

Step 3.9

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

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

Step 3.10

Move the negative in front of the fraction.

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