# Fundamental Rules for Differentiation

### 1. Power Rule:

Rule:

• For a function $f\left(x\right)={x}^{n}$, where $n$ is a constant, the derivative is ${f}^{\mathrm{\prime }}\left(x\right)=n{x}^{n-1}$.

Examples:

• $f\left(x\right)={x}^{3}\text{ }⟹\text{ }{f}^{\mathrm{\prime }}\left(x\right)=3{x}^{2}$
• $g\left(x\right)={x}^{-2}\text{ }⟹\text{ }{g}^{\mathrm{\prime }}\left(x\right)=-2{x}^{-3}$

### 2. Constant Multiple Rule:

Rule:

• If $c$ is a constant and $f\left(x\right)$ is differentiable, then $\frac{d}{dx}\left[cf\left(x\right)\right]=c\cdot \frac{d}{dx}\left[f\left(x\right)\right]$.

Example:

• $f\left(x\right)=5{x}^{2}\text{ }⟹\text{ }{f}^{\mathrm{\prime }}\left(x\right)=5\cdot 2x=10x$

### 3. Sum/Difference Rule:

Rule:

• The derivative of the sum/difference of two functions is the sum/difference of their derivatives: $\frac{d}{dx}\left[f\left(x\right)±g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]±\frac{d}{dx}\left[g\left(x\right)\right]$.

Example:

• $f\left(x\right)={x}^{2}+3x\text{ }⟹\text{ }{f}^{\mathrm{\prime }}\left(x\right)=2x+3$

### 4. Product Rule:

Rule:

• For two functions $u\left(x\right)$ and $v\left(x\right)$, the derivative of their product is: $\frac{d}{dx}\left[u\left(x\right)\cdot v\left(x\right)\right]={u}^{\mathrm{\prime }}\left(x\right)\cdot v\left(x\right)+u\left(x\right)\cdot {v}^{\mathrm{\prime }}\left(x\right)$.

Example:

• $f\left(x\right)={x}^{2}\cdot \mathrm{sin}\left(x\right)\text{ }⟹\text{ }{f}^{\mathrm{\prime }}\left(x\right)=2x\cdot \mathrm{sin}\left(x\right)+{x}^{2}\cdot \mathrm{cos}\left(x\right)$

### 5. Quotient Rule:

Rule:

• For $u\left(x\right)$ and $v\left(x\right)$ where $v\left(x\right)\mathrm{\ne }0$, the derivative of their quotient is: $\frac{d}{dx}\left[\frac{u\left(x\right)}{v\left(x\right)}\right]=\frac{v\left(x\right)\cdot {u}^{\mathrm{\prime }}\left(x\right)-u\left(x\right)\cdot {v}^{\mathrm{\prime }}\left(x\right)}{\left[v\left(x\right){\right]}^{2}}$.

Example:

• $f\left(x\right)=\frac{{x}^{2}}{\mathrm{sin}\left(x\right)}\text{ }⟹\text{ }{f}^{\mathrm{\prime }}\left(x\right)=\frac{\mathrm{sin}\left(x\right)\cdot 2x-{x}^{2}\cdot \mathrm{cos}\left(x\right)}{\left[\mathrm{sin}\left(x\right){\right]}^{2}}$

### 6. Chain Rule:

Rule:

• If $y=f\left(u\right)$ and $u=g\left(x\right)$, then the chain rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$.

Example:

• $f\left(x\right)=\mathrm{sin}\left({x}^{2}\right)\text{ }⟹\text{ }{f}^{\mathrm{\prime }}\left(x\right)=2x\cdot \mathrm{cos}\left({x}^{2}\right)$