# Differentiation from first principle

### Definition of Derivative from First Principles:

The derivative of a function $f\left(x\right)$ at a point $x=a$ is defined as the limit of the difference quotient as the interval between two points approaches zero.

Derivative Definition: The derivative of $f\left(x\right)$ at $x=a$ is given by: ${f}^{\mathrm{\prime }}\left(a\right)={\mathrm{lim}}_{h\to 0}\frac{f\left(a+h\right)-f\left(a\right)}{h}$

### Steps to Compute Derivatives Using First Principles:

1. Find the Difference Quotient:

• Start with the definition of the derivative: $\frac{f\left(a+h\right)-f\left(a\right)}{h}$.
2. Simplify the Expression:

• Expand $f\left(a+h\right)$ and $f\left(a\right)$.
• Subtract $f\left(a\right)$ from $f\left(a+h\right)$.
3. Factor and Simplify:

• Factor out common terms.
• Simplify the expression as much as possible.
4. Take the Limit:

• Let $h$ approach zero ($h\to 0$.
• Evaluate the limit to find the derivative.

### Example:

Let's find the derivative of a simple function $f\left(x\right)={x}^{2}$ from first principles at a point $x=a$.

1. Difference Quotient: ${f}^{\mathrm{\prime }}\left(a\right)={\mathrm{lim}}_{h\to 0}\frac{\left(a+h{\right)}^{2}-{a}^{2}}{h}$

2. Simplify: ${f}^{\mathrm{\prime }}\left(a\right)={\mathrm{lim}}_{h\to 0}\frac{{a}^{2}+2ah+{h}^{2}-{a}^{2}}{h}$ ${f}^{\mathrm{\prime }}\left(a\right)={\mathrm{lim}}_{h\to 0}\frac{2ah+{h}^{2}}{h}$

3. Factor and Simplify: ${f}^{\mathrm{\prime }}\left(a\right)={\mathrm{lim}}_{h\to 0}\frac{h\left(2a+h\right)}{h}$ ${f}^{\mathrm{\prime }}\left(a\right)={\mathrm{lim}}_{h\to 0}\left(2a+h\right)=2a$

4. Take the Limit:

• As $h$ approaches zero, the derivative ${f}^{\mathrm{\prime }}\left(a\right)=2a$.