Calculus Examples

$f (x) = \frac{4}{x^{2}} - 10 x^{3}$

Step 1

By the Sum Rule, the derivative of $\frac{4}{x^{2}} - 10 x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{4}{x^{2}}] + \frac{d}{d x} [- 10 x^{3}]$ .

$\frac{d}{d x} [\frac{4}{x^{2}}] + \frac{d}{d x} [- 10 x^{3}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{4}{x^{2}}]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x^{2}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$4 \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- 10 x^{3}]$

Step 2.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$4 \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{d}{d x} [- 10 x^{3}]$

Step 2.3

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 $g (x) = x^{2}$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$4 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 10 x^{3}]$

Step 2.3.2

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

$4 (- u^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 10 x^{3}]$

Step 2.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$4 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 10 x^{3}]$

Step 2.4

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

$4 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} [- 10 x^{3}]$

Step 2.5

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

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Step 2.5.1

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

$4 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} [- 10 x^{3}]$

Step 2.5.2

Multiply $2$ by $- 2$ .

$4 (- x^{- 4} (2 x)) + \frac{d}{d x} [- 10 x^{3}]$

Step 2.6

Multiply $2$ by $- 1$ .

$4 (- 2 x^{- 4} x) + \frac{d}{d x} [- 10 x^{3}]$

Step 2.7

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

$4 (- 2 (x^{1} x^{- 4})) + \frac{d}{d x} [- 10 x^{3}]$

Step 2.8

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

$4 (- 2 x^{1 - 4}) + \frac{d}{d x} [- 10 x^{3}]$

Step 2.9

Subtract $4$ from $1$ .

$4 (- 2 x^{- 3}) + \frac{d}{d x} [- 10 x^{3}]$

Step 2.10

Multiply $- 2$ by $4$ .

$- 8 x^{- 3} + \frac{d}{d x} [- 10 x^{3}]$

Step 3

Evaluate $\frac{d}{d x} [- 10 x^{3}]$ .

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Step 3.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x^{3}$ with respect to $x$ is $- 10 \frac{d}{d x} [x^{3}]$ .

$- 8 x^{- 3} - 10 \frac{d}{d x} [x^{3}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$- 8 x^{- 3} - 10 (3 x^{2})$

Step 3.3

Multiply $3$ by $- 10$ .

$- 8 x^{- 3} - 30 x^{2}$

Step 4

Simplify.

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Step 4.1

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

$- 8 \frac{1}{x^{3}} - 30 x^{2}$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 8$ and $\frac{1}{x^{3}}$ .

$\frac{- 8}{x^{3}} - 30 x^{2}$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{8}{x^{3}} - 30 x^{2}$

Step 4.3

Reorder terms.

$- 30 x^{2} - \frac{8}{x^{3}}$