Calculus Examples

$f (x) = 4 x^{4} - \frac{1}{4 x^{4}}$

Step 1

By the Sum Rule, the derivative of $4 x^{4} - \frac{1}{4 x^{4}}$ with respect to $x$ is $\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- \frac{1}{4 x^{4}}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [4 x^{4}]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{4}$ with respect to $x$ is $4 \frac{d}{d x} [x^{4}]$ .

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

Step 2.2

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

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

Step 2.3

Multiply $4$ by $4$ .

$16 x^{3} + \frac{d}{d x} [- \frac{1}{4 x^{4}}]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{1}{4 x^{4}}]$ .

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Step 3.1

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{1}{4 x^{4}}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$16 x^{3} - \frac{1}{4} \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.2

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$16 x^{3} - \frac{1}{4} \frac{d}{d x} [{(x^{4})}^{- 1}]$

Step 3.3

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 $g (x) = x^{4}$ .

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Step 3.3.1

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

$16 x^{3} - \frac{1}{4} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4}])$

Step 3.3.2

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

$16 x^{3} - \frac{1}{4} (- u^{- 2} \frac{d}{d x} [x^{4}])$

Step 3.3.3

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

$16 x^{3} - \frac{1}{4} (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}])$

Step 3.4

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

$16 x^{3} - \frac{1}{4} (- {(x^{4})}^{- 2} (4 x^{3}))$

Step 3.5

Multiply the exponents in ${(x^{4})}^{- 2}$ .

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Step 3.5.1

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

$16 x^{3} - \frac{1}{4} (- x^{4 \cdot - 2} (4 x^{3}))$

Step 3.5.2

Multiply $4$ by $- 2$ .

$16 x^{3} - \frac{1}{4} (- x^{- 8} (4 x^{3}))$

Step 3.6

Multiply $4$ by $- 1$ .

$16 x^{3} - \frac{1}{4} (- 4 x^{- 8} x^{3})$

Step 3.7

Multiply $x^{- 8}$ by $x^{3}$ by adding the exponents.

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Step 3.7.1

Move $x^{3}$ .

$16 x^{3} - \frac{1}{4} (- 4 (x^{3} x^{- 8}))$

Step 3.7.2

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

$16 x^{3} - \frac{1}{4} (- 4 x^{3 - 8})$

Step 3.7.3

Subtract $8$ from $3$ .

$16 x^{3} - \frac{1}{4} (- 4 x^{- 5})$

Step 3.8

Multiply $- 4$ by $- 1$ .

$16 x^{3} + 4 (\frac{1}{4}) x^{- 5}$

Step 3.9

Combine $4$ and $\frac{1}{4}$ .

$16 x^{3} + \frac{4}{4} x^{- 5}$

Step 3.10

Combine $\frac{4}{4}$ and $x^{- 5}$ .

$16 x^{3} + \frac{4 x^{- 5}}{4}$

Step 3.11

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$16 x^{3} + \frac{4}{4 x^{5}}$

Step 3.12

Cancel the common factor of $4$ .

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Step 3.12.1

Cancel the common factor.

$16 x^{3} + \frac{4}{4 x^{5}}$

Step 3.12.2

Rewrite the expression.

$16 x^{3} + \frac{1}{x^{5}}$