# Constant Rule

The derivative of f \left( x \right) = c u \left( x \right) is $$f' \left( x \right)= c u' \left( x \right)$$

## Example

Let f \left( x \right) = 2 x^{4}.

It is made of a constant c = 2 and a power function u \left( x \right) = x^{4}.

The function u \left( x \right) = x^{4} is a power function x^{n} with n=4. Apply the power rule \begin{align*} u' \left( x \right) &= 4 x ^{4-1} \\ &= 4 x^{3} \end{align*}

So the derivative of f \left( x \right) is \begin{align*} f' \left( x \right) &= 2 u' \left( x \right) \\ &= 2 \times 4 x ^{3} \\ &= 8 x^{3} \end{align*}

### Question

What is the derivative of f \left( x \right) = 12 x^{ \frac{ 4 }{ 9 } } ?

The function f \left( x \right) = 12 x^{ \frac{ 4 }{ 9 } } is made of a constant c = 12 and a function u \left( x \right) = x^{ \frac{ 4 }{ 9 } }.

The function u \left( x \right) = x^{ \frac{ 4 }{ 9 } } looks like x^{ n }. Use the power rule with n = \frac{ 4 }{ 9 } \begin{align*} u' \left( x \right) &= \frac{ 4 }{ 9 } x^{ \frac{ 4 }{ 9 } - 1 } \\ &= \frac{ 4 }{ 9 } x^{ \frac{ 4 }{ 9 } - \frac{ 9 }{ 9 } } \\ &= \frac{ 4 }{ 9 } x^{ \frac{ -5 }{ 9 } } \end{align*}

Therefore, \begin{align*} f' \left( x \right) &= c u' \left( x \right) \\ &= 12 \times \frac{ 4 }{ 9 } x^{ \frac{ -5 }{ 9 } } \\ &= \frac{ 16 }{ 3 } \times x^{ \frac{ -5 }{ 9 } } \end{align*}