Calculus Examples

$f (x) = \frac{24 x + 10}{x^{2} + 1}$

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 $f (x) = 24 x + 10$ and $g (x) = x^{2} + 1$ .

$\frac{(x^{2} + 1) \frac{d}{d x} [24 x + 10] - (24 x + 10) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $24 x + 10$ with respect to $x$ is $\frac{d}{d x} [24 x] + \frac{d}{d x} [10]$ .

$\frac{(x^{2} + 1) (\frac{d}{d x} [24 x] + \frac{d}{d x} [10]) - (24 x + 10) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 2.2

Since $24$ is constant with respect to $x$ , the derivative of $24 x$ with respect to $x$ is $24 \frac{d}{d x} [x]$ .

$\frac{(x^{2} + 1) (24 \frac{d}{d x} [x] + \frac{d}{d x} [10]) - (24 x + 10) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 2.3

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

$\frac{(x^{2} + 1) (24 \cdot 1 + \frac{d}{d x} [10]) - (24 x + 10) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 2.4

Multiply $24$ by $1$ .

$\frac{(x^{2} + 1) (24 + \frac{d}{d x} [10]) - (24 x + 10) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 2.5

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$\frac{(x^{2} + 1) (24 + 0) - (24 x + 10) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $24$ and $0$ .

$\frac{(x^{2} + 1) \cdot 24 - (24 x + 10) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 2.6.2

Move $24$ to the left of $x^{2} + 1$ .

$\frac{24 \cdot (x^{2} + 1) - (24 x + 10) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 2.7

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

$\frac{24 (x^{2} + 1) - (24 x + 10) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])}{{(x^{2} + 1)}^{2}}$

Step 2.8

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

$\frac{24 (x^{2} + 1) - (24 x + 10) (2 x + \frac{d}{d x} [1])}{{(x^{2} + 1)}^{2}}$

Step 2.9

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

$\frac{24 (x^{2} + 1) - (24 x + 10) (2 x + 0)}{{(x^{2} + 1)}^{2}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

$\frac{24 (x^{2} + 1) - (24 x + 10) (2 x)}{{(x^{2} + 1)}^{2}}$

Step 2.10.2

Multiply $2$ by $- 1$ .

$\frac{24 (x^{2} + 1) - 2 (24 x + 10) x}{{(x^{2} + 1)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{24 x^{2} + 24 \cdot 1 - 2 (24 x + 10) x}{{(x^{2} + 1)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{24 x^{2} + 24 \cdot 1 + (- 2 (24 x) - 2 \cdot 10) x}{{(x^{2} + 1)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{24 x^{2} + 24 \cdot 1 - 2 (24 x) x - 2 \cdot 10 x}{{(x^{2} + 1)}^{2}}$

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 $24$ by $1$ .

$\frac{24 x^{2} + 24 - 2 (24 x) x - 2 \cdot 10 x}{{(x^{2} + 1)}^{2}}$

Step 3.4.1.2

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

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Step 3.4.1.2.1

Move $x$ .

$\frac{24 x^{2} + 24 - 2 (24 (x \cdot x)) - 2 \cdot 10 x}{{(x^{2} + 1)}^{2}}$

Step 3.4.1.2.2

Multiply $x$ by $x$ .

$\frac{24 x^{2} + 24 - 2 (24 x^{2}) - 2 \cdot 10 x}{{(x^{2} + 1)}^{2}}$

Step 3.4.1.3

Multiply $24$ by $- 2$ .

$\frac{24 x^{2} + 24 - 48 x^{2} - 2 \cdot 10 x}{{(x^{2} + 1)}^{2}}$

Step 3.4.1.4

Multiply $- 2$ by $10$ .

$\frac{24 x^{2} + 24 - 48 x^{2} - 20 x}{{(x^{2} + 1)}^{2}}$

Step 3.4.2

Subtract $48 x^{2}$ from $24 x^{2}$ .

$\frac{- 24 x^{2} + 24 - 20 x}{{(x^{2} + 1)}^{2}}$

Step 3.5

Reorder terms.

$\frac{- 24 x^{2} - 20 x + 24}{{(x^{2} + 1)}^{2}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Factor $4$ out of $- 24 x^{2} - 20 x + 24$ .

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Step 3.6.1.1

Factor $4$ out of $- 24 x^{2}$ .

$\frac{4 (- 6 x^{2}) - 20 x + 24}{{(x^{2} + 1)}^{2}}$

Step 3.6.1.2

Factor $4$ out of $- 20 x$ .

$\frac{4 (- 6 x^{2}) + 4 (- 5 x) + 24}{{(x^{2} + 1)}^{2}}$

Step 3.6.1.3

Factor $4$ out of $24$ .

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

Step 3.6.1.4

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

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

Step 3.6.1.5

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

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

Step 3.6.2

Factor by grouping.

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Step 3.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 6 \cdot 6 = - 36$ and whose sum is $b = - 5$ .

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Step 3.6.2.1.1

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

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

Step 3.6.2.1.2

Rewrite $- 5$ as $4$ plus $- 9$

$\frac{4 (- 6 x^{2} + (4 - 9) x + 6)}{{(x^{2} + 1)}^{2}}$

Step 3.6.2.1.3

Apply the distributive property.

$\frac{4 (- 6 x^{2} + 4 x - 9 x + 6)}{{(x^{2} + 1)}^{2}}$

Step 3.6.2.2

Factor out the greatest common factor from each group.

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Step 3.6.2.2.1

Group the first two terms and the last two terms.

$\frac{4 ((- 6 x^{2} + 4 x) - 9 x + 6)}{{(x^{2} + 1)}^{2}}$

Step 3.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- 3 x + 2$ .

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

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

Step 3.7

Factor $- 1$ out of $- 3 x$ .

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

Step 3.8

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

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

Step 3.9

Factor $- 1$ out of $- (3 x) - 1 (- 2)$ .

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

Step 3.10

Rewrite $- (3 x - 2)$ as $- 1 (3 x - 2)$ .

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

Step 3.11

Move the negative in front of the fraction.

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

Step 3.12

Reorder factors in $- \frac{(4 (3 x - 2)) (2 x + 3)}{{(x^{2} + 1)}^{2}}$ .

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