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) = x^{ \frac{ 2 }{ 8 } }` ?
The function `f \left( x \right) = x^{ \frac{ 2 }{ 8 } }` looks like `x^{ n }`. Use the power rule $$ \begin{align*} f' \left( x \right) &= \frac{ 2 }{ 8 } x^{ \frac{ 2 }{ 8 } - 1 } \\ &= \frac{ 2 }{ 8 } x^{ \frac{ 2 }{ 8 } - \frac{ 8 }{ 8 } } \\ &= \frac{ 2 }{ 8 } x^{ \frac{ -3 }{ 4 } } \end{align*} $$