Math Derivative rules Deriving fractions and roots

# Deriving fractions and roots

The simplest way to derive fractions and roots is to apply the power laws first and then the derivation rules.

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### Remember

Fractions can be rewritten as a potency with a negative exponent:
$\frac{1}{a^x}=a^{-x}$

Roots can also be written as a potency with rational exponents:
$\sqrt[n]{a^m}=a^{\frac{m}{n}}$
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### The method

1. Transform the fraction or root into a potency
2. Apply derivation rules
3. If necessary, write potency again as a fraction or root

### Examples

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

1. #### Transform fraction into potency

$f(x)=x^{-2}$
2. #### Apply the power rule

$f'(x)=-2x^{-2-1}=-2x^{-3}$
3. #### Write potency as fraction

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

$f(x)=\sqrt{x^2}$

1. #### Transform root into potency

$f(x)=x^\frac23$
2. #### Apply the pwer rule

$f'(x)=\frac23x^{\frac23-1}=\frac23x^{-\frac13}$
3. #### Rewrite potency

$f'(x)=\frac23\cdot\frac{1}{\sqrt{x}}$ $=\frac{2}{3\sqrt{x}}$
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### Hint

For sums in the root, the chain rule is applied after reshaping.

For sums in the denominator of a fraction, you can also use the chain rule.
Alternatively, if there are more complex fractions, it is recommended to use the quotient rule to avoid the need for reshaping.

### Example

To understand the example, look at the chain rule first

$f(x)=\sqrt{3x^2+3}$

1. #### Transform root into potency

$f(x)=(3x^2+3)^\frac13$
2. #### Apply the power rule

$f'(x)=\frac13(3x^2+3)^{-\frac23}\cdot6x$ $=2x(3x^2+3)^{-\frac23}$
3. #### Rewrite potency

$f'(x)=\frac{2x}{(3x^2+3)^\frac23}$ $=\frac{2x}{\sqrt{(3x^2+3)^2}}$