Calculus Examples

$f (x) = \frac{8 x}{x^{5} + 3}$

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 $f (x) = 8 x$ and $g (x) = x^{5} + 3$ .

$\frac{(x^{5} + 3) \frac{d}{d x} [8 x] - 1 \cdot 8 x \frac{d}{d x} [x^{5} + 3]}{{(x^{5} + 3)}^{2}}$

Step 2

Differentiate.

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\frac{(x^{5} + 3) (8 \frac{d}{d x} [x]) - 1 \cdot 8 x \frac{d}{d x} [x^{5} + 3]}{{(x^{5} + 3)}^{2}}$

Step 2.2

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

$\frac{(x^{5} + 3) (8 \cdot 1) - 1 \cdot 8 x \frac{d}{d x} [x^{5} + 3]}{{(x^{5} + 3)}^{2}}$

Step 2.3

Multiply $8$ by $1$ .

$\frac{(x^{5} + 3) \cdot 8 - 1 \cdot 8 x \frac{d}{d x} [x^{5} + 3]}{{(x^{5} + 3)}^{2}}$

Step 2.4

By the Sum Rule, the derivative of $x^{5} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [3]$ .

$\frac{(x^{5} + 3) \cdot 8 - 1 \cdot 8 x (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [3])}{{(x^{5} + 3)}^{2}}$

Step 2.5

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

$\frac{(x^{5} + 3) \cdot 8 - 1 \cdot 8 x (5 x^{4} + \frac{d}{d x} [3])}{{(x^{5} + 3)}^{2}}$

Step 2.6

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{(x^{5} + 3) \cdot 8 - 1 \cdot 8 x (5 x^{4} + 0)}{{(x^{5} + 3)}^{2}}$

Step 2.7

Add $5 x^{4}$ and $0$ .

$\frac{(x^{5} + 3) \cdot 8 - 1 \cdot 8 x (5 x^{4})}{{(x^{5} + 3)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{x^{5} \cdot 8 + 3 \cdot 8 - 1 \cdot 8 x (5 x^{4})}{{(x^{5} + 3)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Move $8$ to the left of $x^{5}$ .

$\frac{8 \cdot x^{5} + 3 \cdot 8 - 1 \cdot 8 x (5 x^{4})}{{(x^{5} + 3)}^{2}}$

Step 3.2.1.2

Multiply $3$ by $8$ .

$\frac{8 x^{5} + 24 - 1 \cdot 8 x (5 x^{4})}{{(x^{5} + 3)}^{2}}$

Step 3.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{8 x^{5} + 24 - 1 \cdot 8 \cdot 5 x \cdot x^{4}}{{(x^{5} + 3)}^{2}}$

Step 3.2.1.4

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 3.2.1.4.1

Move $x^{4}$ .

$\frac{8 x^{5} + 24 - 1 \cdot 8 \cdot 5 (x^{4} x)}{{(x^{5} + 3)}^{2}}$

Step 3.2.1.4.2

Multiply $x^{4}$ by $x$ .

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Step 3.2.1.4.2.1

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

$\frac{8 x^{5} + 24 - 1 \cdot 8 \cdot 5 (x^{4} x^{1})}{{(x^{5} + 3)}^{2}}$

Step 3.2.1.4.2.2

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

$\frac{8 x^{5} + 24 - 1 \cdot 8 \cdot 5 x^{4 + 1}}{{(x^{5} + 3)}^{2}}$

Step 3.2.1.4.3

Add $4$ and $1$ .

$\frac{8 x^{5} + 24 - 1 \cdot 8 \cdot 5 x^{5}}{{(x^{5} + 3)}^{2}}$

Step 3.2.1.5

Multiply $- 1 \cdot 8 \cdot 5$ .

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Step 3.2.1.5.1

Multiply $- 1$ by $8$ .

$\frac{8 x^{5} + 24 - 8 \cdot 5 x^{5}}{{(x^{5} + 3)}^{2}}$

Step 3.2.1.5.2

Multiply $- 8$ by $5$ .

$\frac{8 x^{5} + 24 - 40 x^{5}}{{(x^{5} + 3)}^{2}}$

Step 3.2.2

Subtract $40 x^{5}$ from $8 x^{5}$ .

$\frac{- 32 x^{5} + 24}{{(x^{5} + 3)}^{2}}$

Step 3.3

Factor $8$ out of $- 32 x^{5} + 24$ .

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Step 3.3.1

Factor $8$ out of $- 32 x^{5}$ .

$\frac{8 (- 4 x^{5}) + 24}{{(x^{5} + 3)}^{2}}$

Step 3.3.2

Factor $8$ out of $24$ .

$\frac{8 (- 4 x^{5}) + 8 (3)}{{(x^{5} + 3)}^{2}}$

Step 3.3.3

Factor $8$ out of $8 (- 4 x^{5}) + 8 (3)$ .

$\frac{8 (- 4 x^{5} + 3)}{{(x^{5} + 3)}^{2}}$

Step 3.4

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

$\frac{8 (- (4 x^{5}) + 3)}{{(x^{5} + 3)}^{2}}$

Step 3.5

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

$\frac{8 (- (4 x^{5}) - 1 (- 3))}{{(x^{5} + 3)}^{2}}$

Step 3.6

Factor $- 1$ out of $- (4 x^{5}) - 1 (- 3)$ .

$\frac{8 (- (4 x^{5} - 3))}{{(x^{5} + 3)}^{2}}$

Step 3.7

Rewrite $- (4 x^{5} - 3)$ as $- 1 (4 x^{5} - 3)$ .

$\frac{8 (- 1 (4 x^{5} - 3))}{{(x^{5} + 3)}^{2}}$

Step 3.8

Move the negative in front of the fraction.

$- \frac{8 (4 x^{5} - 3)}{{(x^{5} + 3)}^{2}}$