# 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) = \frac{ 7 }{ 9 } x^{ 844 } ?

The function f \left( x \right) = \frac{ 7 }{ 9 } x^{ 844 } is made of a constant c = \frac{ 7 }{ 9 } and a function u \left( x \right) = x^{ 844 }.

The function u \left( x \right) = x^{ 844 } looks like x^{ n }. Use the power rule with n = 844 \begin{align*} u' \left( x \right) &= 844 x^{ 844 - 1 } \\ &= 844 x^{ 843 } \end{align*}

Therefore, \begin{align*} f' \left( x \right) &= c u' \left( x \right) \\ &= \frac{ 7 }{ 9 } \times 844 x^{ 843 } \\ &= \frac{ 5908 }{ 9 } \times x^{ 843 } \end{align*}