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:
$\sqrt[n]{a^m}=a^{\frac{m}{n}}$
$\frac{1}{a^x}=a^{-x}$
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
- Transform the fraction or root into a potency
- Apply derivation rules
- If necessary, write potency again as a fraction or root
Examples
$f(x)=\frac{1}{x^2}$
Transform fraction into potency
$f(x)=x^{-2}$Apply the power rule
$f'(x)=-2x^{-2-1}=-2x^{-3}$Write potency as fraction
$f'(x)=-\frac{2}{x^3}$
$f(x)=\sqrt[3]{x^2}$
Transform root into potency
$f(x)=x^\frac23$Apply the pwer rule
$f'(x)=\frac23x^{\frac23-1}=\frac23x^{-\frac13}$Rewrite potency
$f'(x)=\frac23\cdot\frac{1}{\sqrt[3]{x}}$ $=\frac{2}{3\sqrt[3]{x}}$
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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.
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
$f(x)=\sqrt[3]{3x^2+3}$
Transform root into potency
$f(x)=(3x^2+3)^\frac13$Apply the power rule
$f'(x)=\frac13(3x^2+3)^{-\frac23}\cdot6x$ $=2x(3x^2+3)^{-\frac23}$Rewrite potency
$f'(x)=\frac{2x}{(3x^2+3)^\frac23}$ $=\frac{2x}{\sqrt[3]{(3x^2+3)^2}}$