Calculus Examples

$g (x) = \frac{x^{2} + 1}{x^{5}}$

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

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

Step 2

Differentiate.

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Step 2.1

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 2.1.1

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

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

Step 2.1.2

Multiply $5$ by $2$ .

$\frac{x^{5} \frac{d}{d x} [x^{2} + 1] - (x^{2} + 1) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.2

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

$\frac{x^{5} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) - (x^{2} + 1) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.3

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

$\frac{x^{5} (2 x + \frac{d}{d x} [1]) - (x^{2} + 1) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.4

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

$\frac{x^{5} (2 x + 0) - (x^{2} + 1) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.5

Add $2 x$ and $0$ .

$\frac{x^{5} (2 x) - (x^{2} + 1) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 3.1

Move $x$ .

$\frac{x \cdot x^{5} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.2

Multiply $x$ by $x^{5}$ .

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Step 3.2.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{5} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 5} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.3

Add $1$ and $5$ .

$\frac{x^{6} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4

Move $2$ to the left of $x^{6}$ .

$\frac{2 \cdot x^{6} - (x^{2} + 1) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{2 x^{6} - (x^{2} + 1) (5 x^{4})}{x^{10}}$

Step 6

Simplify with factoring out.

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Step 6.1

Multiply $5$ by $- 1$ .

$\frac{2 x^{6} - 5 (x^{2} + 1) x^{4}}{x^{10}}$

Step 6.2

Factor $x^{4}$ out of $2 x^{6} - 5 (x^{2} + 1) x^{4}$ .

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Step 6.2.1

Factor $x^{4}$ out of $2 x^{6}$ .

$\frac{x^{4} (2 x^{2}) - 5 (x^{2} + 1) x^{4}}{x^{10}}$

Step 6.2.2

Factor $x^{4}$ out of $- 5 (x^{2} + 1) x^{4}$ .

$\frac{x^{4} (2 x^{2}) + x^{4} (- 5 (x^{2} + 1))}{x^{10}}$

Step 6.2.3

Factor $x^{4}$ out of $x^{4} (2 x^{2}) + x^{4} (- 5 (x^{2} + 1))$ .

$\frac{x^{4} (2 x^{2} - 5 (x^{2} + 1))}{x^{10}}$

Step 7

Cancel the common factors.

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Step 7.1

Factor $x^{4}$ out of $x^{10}$ .

$\frac{x^{4} (2 x^{2} - 5 (x^{2} + 1))}{x^{4} x^{6}}$

Step 7.2

Cancel the common factor.

$\frac{x^{4} (2 x^{2} - 5 (x^{2} + 1))}{x^{4} x^{6}}$

Step 7.3

Rewrite the expression.

$\frac{2 x^{2} - 5 (x^{2} + 1)}{x^{6}}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{2 x^{2} - 5 x^{2} - 5 \cdot 1}{x^{6}}$

Step 8.2

Simplify the numerator.

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Step 8.2.1

Multiply $- 5$ by $1$ .

$\frac{2 x^{2} - 5 x^{2} - 5}{x^{6}}$

Step 8.2.2

Subtract $5 x^{2}$ from $2 x^{2}$ .

$\frac{- 3 x^{2} - 5}{x^{6}}$

Step 8.3

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

$\frac{- (3 x^{2}) - 5}{x^{6}}$

Step 8.4

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

$\frac{- (3 x^{2}) - 1 (5)}{x^{6}}$

Step 8.5

Factor $- 1$ out of $- (3 x^{2}) - 1 (5)$ .

$\frac{- (3 x^{2} + 5)}{x^{6}}$

Step 8.6

Rewrite $- (3 x^{2} + 5)$ as $- 1 (3 x^{2} + 5)$ .

$\frac{- 1 (3 x^{2} + 5)}{x^{6}}$

Step 8.7

Move the negative in front of the fraction.

$- \frac{3 x^{2} + 5}{x^{6}}$