Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $5$ by $1$ .

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

Step 2.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $5$ and $0$ .

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

Step 2.6.2

Move $5$ to the left of $x^{2} + 1$ .

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

$\frac{5 (x^{2} + 1) - (5 x - 2) (2 x + 0)}{{(x^{2} + 1)}^{2}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

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

Step 2.10.2

Multiply $2$ by $- 1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $5$ by $1$ .

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

Step 3.4.1.2

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

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Step 3.4.1.2.1

Move $x$ .

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

Step 3.4.1.2.2

Multiply $x$ by $x$ .

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

Step 3.4.1.3

Multiply $5$ by $- 2$ .

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

Step 3.4.1.4

Multiply $- 2$ by $- 2$ .

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

Step 3.4.2

Subtract $10 x^{2}$ from $5 x^{2}$ .

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

Step 3.5

Reorder terms.

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

Step 3.6

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

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

Step 3.7

Factor $- 1$ out of $4 x$ .

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Rewrite $- (5 x^{2} - 4 x - 5)$ as $- 1 (5 x^{2} - 4 x - 5)$ .

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

Step 3.12

Move the negative in front of the fraction.

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