Power Rule

102. Power Rule

\begin{aligned} 𝑓 (𝑥) & = 𝑥^{𝑛}, 𝑛 \neq 0 \\ 𝑓^{'} (𝑥) & = 𝑛 𝑥^{𝑛 - 1} \end{aligned}

Example

\begin{aligned} 𝑓 (𝑥) & = 𝑥^{3} \\ 𝑓^{'} (𝑥) & = 3 𝑥^{2} \end{aligned}

Example

\begin{aligned} \frac{𝑑}{𝑑 𝑥} (\sqrt[3]{𝑥^{2}}) & = \frac{𝑑}{𝑑 𝑥} ({(𝑥^{2})}^{\frac{1}{3}}) \\ = \frac{𝑑}{𝑑 𝑥} (𝑥^{2 \times \frac{1}{3}}) \\ = \frac{𝑑}{𝑑 𝑥} (𝑥^{\frac{2}{3}}) \\ = \frac{𝑑}{𝑑 𝑥} (\frac{2}{3} 𝑥^{- \frac{1}{3}}) \end{aligned}

102.0.1. Constant Rule

\frac{𝑑}{𝑑 𝑥} [𝑘] = 0

Example

\frac{𝑑}{𝑑 𝑥} [- 3] = 0

102.0.2. Constant Multiple Rule

\begin{aligned} \frac{𝑑}{𝑑 𝑥} [𝑘 𝑓 (𝑥)] & = 𝑘 \frac{𝑑}{𝑑 𝑥} [𝑓 (𝑥)] \\ = 𝑘 𝑓^{'} (𝑥) \end{aligned}

Example

\begin{aligned} \frac{𝑑}{𝑑 𝑥} [2 𝑥^{5}] & = 2 \frac{𝑑}{𝑑 𝑥} [𝑥^{5}] \\ = 2 \cdot 5 𝑥^{4} \\ = 10 𝑥^{4} \end{aligned}

102.0.3. Sum Rule

\begin{aligned} \frac{𝑑}{𝑑 𝑥} [𝑓 (𝑥) + 𝑔 (𝑥)] & = \frac{𝑑}{𝑑 𝑥} [𝑓 (𝑥)] + \frac{𝑑}{𝑑 𝑥} [𝑔 (𝑥)] \\ = 𝑓^{'} (𝑥) + 𝑔^{'} (𝑥) \end{aligned}

Example

\begin{aligned} \frac{𝑑}{𝑑 𝑥} [𝑥^{3} + 𝑥^{- 4}] & = \frac{𝑑}{𝑑 𝑥} [𝑥^{3}] + \frac{𝑑}{𝑑 𝑥} [𝑥^{- 4}] \\ = 3 𝑥^{2} + (- 4 𝑥^{- 5}) \\ = 3 𝑥^{2} - 4 𝑥^{- 5} \end{aligned}

102.0.4. Difference Rule

\begin{aligned} \frac{𝑑}{𝑑 𝑥} [𝑓 (𝑥) - 𝑔 (𝑥)] & = \frac{𝑑}{𝑑 𝑥} [𝑓 (𝑥)] - \frac{𝑑}{𝑑 𝑥} [𝑔 (𝑥)] \\ = 𝑓^{'} (𝑥) - 𝑔^{'} (𝑥) \end{aligned}

Example

\begin{aligned} \frac{𝑑}{𝑑 𝑥} [𝑥^{4} - 𝑥^{3}] & = \frac{𝑑}{𝑑 𝑥} [𝑥^{4}] - \frac{𝑑}{𝑑 𝑥} [𝑥^{3}] \\ = 4 𝑥^{3} - 3 𝑥^{2} \end{aligned}

102.0.5. Square Root

Example

\begin{aligned} \frac{𝑑}{𝑑 𝑥} \sqrt[4]{𝑥} & = \frac{𝑑}{𝑑 𝑥} 𝑥^{\frac{1}{4}} \\ = \frac{1}{4} 𝑥^{\frac{1}{4} - 1} \\ = \frac{1}{4} \cdot 𝑥^{- \frac{3}{4}} \\ = \frac{1}{4} \cdot \frac{1}{𝑥^{3 / 4}} \\ = \frac{1}{4 𝑥^{3 / 4}} \end{aligned}

102.0.6. Derivative of a Polynomial

Example

\begin{matrix} 𝑓 (𝑥) = 2 𝑥^{3} - 7 𝑥^{2} + 3 𝑥 - 100 \\ 𝑓^{'} (𝑥) = 2 \cdot 3 𝑥^{2} - 7 \cdot 2 𝑥 + 3 + 0 \end{matrix}

Example

ℎ (𝑥) = 3 𝑓 (𝑥) + 2 𝑔 (𝑥)

Evaluate $\frac{𝑑}{𝑑 𝑥} ℎ (𝑥)$ at $𝑥 = 9$

\begin{aligned} \frac{𝑑}{𝑑 𝑥} (ℎ (𝑥)) & = \frac{𝑑}{𝑑 𝑥} (3 𝑓 (𝑥) + 2 𝑔 (𝑥)) \\ = \frac{𝑑}{𝑑 𝑥} 3 𝑓 (𝑥) + \frac{𝑑}{𝑑 𝑥} 2 𝑔 (𝑥) \\ = 3 \frac{𝑑}{𝑑 𝑥} 𝑓 (𝑥) + 2 \frac{𝑑}{𝑑 𝑥} 𝑔 (𝑥) \end{aligned}

Evaluate $ℎ^{'} (9)$

ℎ^{'} (9) = 3 𝑓^{'} (9) + 2 𝑔^{'} (9)

Example

\begin{aligned} 𝑔 (𝑥) & = \frac{2}{𝑥^{3}} - \frac{1}{𝑥^{2}} \\ \frac{𝑑}{𝑑 𝑥} (𝑔 (𝑥)) & = \frac{𝑑}{𝑑 𝑥} (2 𝑥^{- 3} - 1 𝑥^{- 2}) \\ 𝑔^{'} (𝑥) & = 2 \cdot (- 3) 𝑥^{- 4} - (- 2) 𝑥^{- 3} \\ = - 6 𝑥^{- 4} + 2 𝑥^{- 3} \end{aligned}

\begin{aligned} 𝑔^{'} (2) & = - 6 \cdot 2^{- 4} + 2 \cdot 2^{- 3} \\ = - \frac{6}{2^{4}} + \frac{2}{2^{3}} \\ = - \frac{3}{8} + \frac{2}{8} \\ = - \frac{1}{8} \end{aligned}

Example

−9 −8

\begin{matrix} 𝑓 (𝑥) = 𝑥^{3} - 6 𝑥^{2} + 𝑥 - 5 \\ 𝑦 = 𝑚 𝑥 + 𝑏 \\ 𝑓^{'} (1) = −8 𝑦 = −8 𝑥 + 𝑏 \\ - 9 = −8 \cdot 1 + 𝑏 \\ - 9 = −8 + 𝑏 \\ - 9 + 8 = −8 + 8 + 𝑏 \\ 𝑏 = - 1 \end{matrix}

Tangent line to $𝑓 (𝑥)$ at $𝑥 = 1$

𝑦 = - 8 𝑥 - 1

102.0.7. Sin

\frac{𝑑}{𝑑 𝑥} sin (𝑥) = cos (𝑥)

102.0.8. Cos

\frac{𝑑}{𝑑 𝑥} cos (𝑥) = - sin (𝑥)

102.0.9. $𝒆^{𝒙}$

\frac{𝑑}{𝑑 𝑥} 𝑒^{𝑥} = 𝑒^{𝑥}

102.0.10. $𝐥𝐧 (𝒙)$

ln (𝑥) = \frac{1}{𝑥}