Calculus Examples

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

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $- 3$ by $1$ .

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

Step 2.7

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

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

Step 2.8

Add $2 x - 3$ and $0$ .

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

Step 2.9

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

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

Step 2.10

Simplify with factoring out.

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Step 2.10.1

Multiply $2$ by $- 1$ .

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

Step 2.10.2

Factor $x$ out of $x^{2} (2 x - 3) - 2 (x^{2} - 3 x + 1) x$ .

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Step 2.10.2.1

Factor $x$ out of $x^{2} (2 x - 3)$ .

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

Step 2.10.2.2

Factor $x$ out of $- 2 (x^{2} - 3 x + 1) x$ .

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

Step 2.10.2.3

Factor $x$ out of $x (x (2 x - 3)) + x (- 2 (x^{2} - 3 x + 1))$ .

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

Step 3

Cancel the common factors.

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Step 3.1

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

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.2

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

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Step 4.3.1.2.1

Move $x$ .

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

Step 4.3.1.2.2

Multiply $x$ by $x$ .

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

Step 4.3.1.3

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

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

Step 4.3.1.4

Multiply $- 3$ by $- 2$ .

$\frac{2 x^{2} - 3 x - 2 x^{2} + 6 x - 2 \cdot 1}{x^{3}}$

Step 4.3.1.5

Multiply $- 2$ by $1$ .

$\frac{2 x^{2} - 3 x - 2 x^{2} + 6 x - 2}{x^{3}}$

Step 4.3.2

Combine the opposite terms in $2 x^{2} - 3 x - 2 x^{2} + 6 x - 2$ .

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Step 4.3.2.1

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

$\frac{- 3 x + 0 + 6 x - 2}{x^{3}}$

Step 4.3.2.2

Add $- 3 x$ and $0$ .

$\frac{- 3 x + 6 x - 2}{x^{3}}$

Step 4.3.3

Add $- 3 x$ and $6 x$ .

$\frac{3 x - 2}{x^{3}}$