Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{4} + 3 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{4} + 3 x$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4} + 3 x]$

Step 1.2

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

$- u^{- 2} \frac{d}{d x} [x^{4} + 3 x]$

Step 1.3

Replace all occurrences of $u$ with $x^{4} + 3 x$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{4} + 3 x$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [3 x]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 2.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 2.4

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

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

Step 2.5

Multiply $3$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2

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

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Multiply $4$ by $- 1$ .

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

Step 3.5

Multiply $- 1$ by $3$ .

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

Step 3.6

Simplify the denominator.

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Step 3.6.1

Factor $x$ out of $x^{4} + 3 x$ .

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Step 3.6.1.1

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

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

Step 3.6.1.2

Factor $x$ out of $3 x$ .

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

Step 3.6.1.3

Factor $x$ out of $x \cdot x^{3} + x \cdot 3$ .

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

Step 3.6.2

Apply the product rule to $x (x^{3} + 3)$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

Move the negative in front of the fraction.

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