# Differentiating of algebraic functions

### 1. Derivatives of Polynomial Functions:

• Constant Function:

• $\frac{d}{dx}\left[c\right]=0$ where $c$ is a constant.
• Linear Function:

• $\frac{d}{dx}\left[ax+b\right]=a$ where $a$ and $b$ are constants.

• $\frac{d}{dx}\left[a{x}^{2}+bx+c\right]=2ax+b$

### 2. Derivatives of Exponential and Logarithmic Functions:

• Exponential Function:

• $\frac{d}{dx}\left[{e}^{x}\right]={e}^{x}$
• $\frac{d}{dx}\left[{a}^{x}\right]={a}^{x}\cdot \mathrm{ln}\left(a\right)$ where $a>0$
• Natural Logarithm Function:

• $\frac{d}{dx}\left[\mathrm{ln}\left(x\right)\right]=\frac{1}{x}$

### 3. Derivatives of Radical Functions:

• Square Root Function:

• $\frac{d}{dx}\left[\sqrt{x}\right]=\frac{1}{2\sqrt{x}}$

• Higher Order Roots:

• $\frac{d}{dx}\left[{x}^{n}\right]=n{x}^{n-1}$ where $n$ is a constant.

### 4. Derivatives of Rational Functions:

• Constant Divided by a Function:

• $\frac{d}{dx}\left[\frac{c}{f\left(x\right)}\right]=-\frac{c\cdot {f}^{\mathrm{\prime }}\left(x\right)}{\left[f\left(x\right){\right]}^{2}}$
• Quotient Rule for Functions:

• $\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{g\left(x\right)\cdot {f}^{\mathrm{\prime }}\left(x\right)-f\left(x\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)}{\left[g\left(x\right){\right]}^{2}}$

### 5. Chain Rule for Algebraic Functions:

• Chain Rule Application:
• When functions are nested or composed, the chain rule is applied.
• Example: $f\left(x\right)={e}^{3{x}^{2}}$
• ${f}^{\mathrm{\prime }}\left(x\right)=6x\cdot {e}^{3{x}^{2}}$