Calculus Examples

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

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 $f (x) = 2 x^{2} - 3 x + 1$ and $g (x) = x$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $2 x^{2} - 3 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $2$ .

$\frac{x (4 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]) - (2 x^{2} - 3 x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 2.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$\frac{x (4 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [1]) - (2 x^{2} - 3 x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 2.6

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

$\frac{x (4 x - 3 \cdot 1 + \frac{d}{d x} [1]) - (2 x^{2} - 3 x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 2.7

Multiply $- 3$ by $1$ .

$\frac{x (4 x - 3 + \frac{d}{d x} [1]) - (2 x^{2} - 3 x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 2.8

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

$\frac{x (4 x - 3 + 0) - (2 x^{2} - 3 x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 2.9

Add $4 x - 3$ and $0$ .

$\frac{x (4 x - 3) - (2 x^{2} - 3 x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 2.10

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

$\frac{x (4 x - 3) - (2 x^{2} - 3 x + 1) \cdot 1}{x^{2}}$

Step 2.11

Multiply $- 1$ by $1$ .

$\frac{x (4 x - 3) - (2 x^{2} - 3 x + 1)}{x^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{x (4 x) + x \cdot - 3 - (2 x^{2} - 3 x + 1)}{x^{2}}$

Step 3.2

Apply the distributive property.

$\frac{x (4 x) + x \cdot - 3 - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}}$

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

Rewrite using the commutative property of multiplication.

$\frac{4 x \cdot x + x \cdot - 3 - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}}$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

$\frac{4 (x \cdot x) + x \cdot - 3 - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}}$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + x \cdot - 3 - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}}$

Step 3.3.1.3

Move $- 3$ to the left of $x$ .

$\frac{4 x^{2} - 3 \cdot x - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}}$

Step 3.3.1.4

Multiply $2$ by $- 1$ .

$\frac{4 x^{2} - 3 x - 2 x^{2} - (- 3 x) - 1 \cdot 1}{x^{2}}$

Step 3.3.1.5

Multiply $- 3$ by $- 1$ .

$\frac{4 x^{2} - 3 x - 2 x^{2} + 3 x - 1 \cdot 1}{x^{2}}$

Step 3.3.1.6

Multiply $- 1$ by $1$ .

$\frac{4 x^{2} - 3 x - 2 x^{2} + 3 x - 1}{x^{2}}$

Step 3.3.2

Combine the opposite terms in $4 x^{2} - 3 x - 2 x^{2} + 3 x - 1$ .

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Step 3.3.2.1

Add $- 3 x$ and $3 x$ .

$\frac{4 x^{2} - 2 x^{2} + 0 - 1}{x^{2}}$

Step 3.3.2.2

Add $4 x^{2} - 2 x^{2}$ and $0$ .

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

Step 3.3.3

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

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