Calculus Examples

$\frac{x^{2} + 5 x - 1}{x^{2}}$

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 $f (x) = x^{2} + 5 x - 1$ and $g (x) = x^{2}$ .

$\frac{x^{2} \frac{d}{d x} [x^{2} + 5 x - 1] - (x^{2} + 5 x - 1) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

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

$\frac{x^{2} \frac{d}{d x} [x^{2} + 5 x - 1] - (x^{2} + 5 x - 1) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 2.1.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [x^{2} + 5 x - 1] - (x^{2} + 5 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2

By the Sum Rule, the derivative of $x^{2} + 5 x - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 1]$ .

$\frac{x^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 1]) - (x^{2} + 5 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3

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

$\frac{x^{2} (2 x + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 1]) - (x^{2} + 5 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{x^{2} (2 x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (x^{2} + 5 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5

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

$\frac{x^{2} (2 x + 5 \cdot 1 + \frac{d}{d x} [- 1]) - (x^{2} + 5 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6

Multiply $5$ by $1$ .

$\frac{x^{2} (2 x + 5 + \frac{d}{d x} [- 1]) - (x^{2} + 5 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.7

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

$\frac{x^{2} (2 x + 5 + 0) - (x^{2} + 5 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.8

Add $2 x + 5$ and $0$ .

$\frac{x^{2} (2 x + 5) - (x^{2} + 5 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.9

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

$\frac{x^{2} (2 x + 5) - (x^{2} + 5 x - 1) (2 x)}{x^{4}}$

Step 2.10

Simplify with factoring out.

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Step 2.10.1

Multiply $2$ by $- 1$ .

$\frac{x^{2} (2 x + 5) - 2 (x^{2} + 5 x - 1) x}{x^{4}}$

Step 2.10.2

Factor $x$ out of $x^{2} (2 x + 5) - 2 (x^{2} + 5 x - 1) x$ .

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Step 2.10.2.1

Factor $x$ out of $x^{2} (2 x + 5)$ .

$\frac{x (x (2 x + 5)) - 2 (x^{2} + 5 x - 1) x}{x^{4}}$

Step 2.10.2.2

Factor $x$ out of $- 2 (x^{2} + 5 x - 1) x$ .

$\frac{x (x (2 x + 5)) + x (- 2 (x^{2} + 5 x - 1))}{x^{4}}$

Step 2.10.2.3

Factor $x$ out of $x (x (2 x + 5)) + x (- 2 (x^{2} + 5 x - 1))$ .

$\frac{x (x (2 x + 5) - 2 (x^{2} + 5 x - 1))}{x^{4}}$

Step 3

Cancel the common factors.

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Step 3.1

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

$\frac{x (x (2 x + 5) - 2 (x^{2} + 5 x - 1))}{x \cdot x^{3}}$

Step 3.2

Cancel the common factor.

$\frac{x (x (2 x + 5) - 2 (x^{2} + 5 x - 1))}{x \cdot x^{3}}$

Step 3.3

Rewrite the expression.

$\frac{x (2 x + 5) - 2 (x^{2} + 5 x - 1)}{x^{3}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{x (2 x) + x \cdot 5 - 2 (x^{2} + 5 x - 1)}{x^{3}}$

Step 4.2

Apply the distributive property.

$\frac{x (2 x) + x \cdot 5 - 2 x^{2} - 2 (5 x) - 2 \cdot - 1}{x^{3}}$

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.

$\frac{2 x \cdot x + x \cdot 5 - 2 x^{2} - 2 (5 x) - 2 \cdot - 1}{x^{3}}$

Step 4.3.1.2

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

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Step 4.3.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot 5 - 2 x^{2} - 2 (5 x) - 2 \cdot - 1}{x^{3}}$

Step 4.3.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot 5 - 2 x^{2} - 2 (5 x) - 2 \cdot - 1}{x^{3}}$

Step 4.3.1.3

Move $5$ to the left of $x$ .

$\frac{2 x^{2} + 5 \cdot x - 2 x^{2} - 2 (5 x) - 2 \cdot - 1}{x^{3}}$

Step 4.3.1.4

Multiply $5$ by $- 2$ .

$\frac{2 x^{2} + 5 x - 2 x^{2} - 10 x - 2 \cdot - 1}{x^{3}}$

Step 4.3.1.5

Multiply $- 2$ by $- 1$ .

$\frac{2 x^{2} + 5 x - 2 x^{2} - 10 x + 2}{x^{3}}$

Step 4.3.2

Combine the opposite terms in $2 x^{2} + 5 x - 2 x^{2} - 10 x + 2$ .

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Step 4.3.2.1

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

$\frac{5 x + 0 - 10 x + 2}{x^{3}}$

Step 4.3.2.2

Add $5 x$ and $0$ .

$\frac{5 x - 10 x + 2}{x^{3}}$

Step 4.3.3

Subtract $10 x$ from $5 x$ .

$\frac{- 5 x + 2}{x^{3}}$

Step 4.4

Factor $- 1$ out of $- 5 x$ .

$\frac{- (5 x) + 2}{x^{3}}$

Step 4.5

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (5 x) - 1 (- 2)}{x^{3}}$

Step 4.6

Factor $- 1$ out of $- (5 x) - 1 (- 2)$ .

$\frac{- (5 x - 2)}{x^{3}}$

Step 4.7

Rewrite $- (5 x - 2)$ as $- 1 (5 x - 2)$ .

$\frac{- 1 (5 x - 2)}{x^{3}}$

Step 4.8

Move the negative in front of the fraction.

$- \frac{5 x - 2}{x^{3}}$