# Elementary Derivatives

This lesson covers elementary derivatives in Economics.

## Example

### Constant

The derivative of a constant f \left( x \right) = c is f' \left( x \right) = 0.

For example,

Here c = 0, so \begin{align*} f' \left( x \right) = 0 \end{align*}

### Power Rule

The derivative of a power f \left( x \right) = x^{n} is f' \left( x \right) = n x^{n-1}.

For example,

Here n = \frac{1}{3}, so \begin{align*} f' \left( x \right) &= \frac{1}{3} x^{\frac{1}{3} - 1} \\ &= \frac{1}{3} x^{\frac{1}{3} - \frac{3}{3}} \\ &= \frac{1}{3} x^{- \frac{2}{3}} \end{align*}

Remark: The power rule encompasses rules you may have learned.

Rewrite \begin{align*} f' \left( x \right) &= x \\ &= x^{1} \end{align*} Apply the power rule with n = 1 \begin{align*} f' \left( x \right) &= 1 \times x^{1-1} \\ &= x^{0} \\ &= 1 \end{align*}

Rewrite \begin{align*} f' \left( x \right) &= \frac{1}{x} \\ &= x^{-1} \end{align*} Apply the power rule with n = -1 \begin{align*} f' \left( x \right) &= -1 x^{-1-1} \\ &= - x^{-2} \\ &= - \frac{1}{x^2} \end{align*}

Rewrite \begin{align*} f' \left( x \right) &= \sqrt{x} \\ &= x^{\frac{1}{2}} \end{align*} Apply the power rule with n = \frac{1}{2} \begin{align*} f' \left( x \right) &= \frac{1}{2} x^{\frac{1}{2}-1} \\ &= \frac{1}{2} x^{\frac{1}{2}-\frac{2}{2}} \\ &= \frac{1}{2} x^{-\frac{1}{2}} \\ &= \frac{1}{2 x^{\frac{1}{2}}} \\ &= \frac{1}{2 \sqrt{x}} \end{align*}

### Logarithm

The derivative of the logarithm f \left( x \right) = \ln \left( x \right) is f' \left( x \right) = \frac{1}{x}.

### Exponential

The derivative of the exponential f \left( x \right) = e^{x} is f' \left( x \right) = e^{x}.

### Question

What is the derivative of f \left( x \right) = \frac{ 3 }{ 6 } ?

The function f \left( x \right) = \frac{ 3 }{ 6 } is constant. So its derivative is \begin{align*} f' \left( x \right) &= 0 \end{align*}