# Differentiation from First Principles

## Differentiation from First Principles

Consider a function $y=f(x)$ where $\color{teal}A$ is the point $(x,f(x))$ and $\color{teal}B$ is the point $(x+h,f(x+h))$. The chord $\color{teal}A\color{teal}B$ has gradient $\dfrac{f(x+h)-f(x)}{(x+h)-x} = \dfrac{f(x+h)-f(x)}{h}$. The gradient of the tangent at the variable point $(x,f(x))$ is the limiting value of $\dfrac{f(x+h)-f(x)}{h}$ as $h$ approaches $0$. This formula gives the gradient of the tangent to the curve $y=f(x)$ at the point $(x,f(x))$, for any value of the variable $x$ for which this limit exists. Since there is at most one value of the gradient for each value of $x$, the formula is actually a function.

The derivative of $y=f(x)$ is defined as; $$f'(x)=\lim_{h \rightarrow 0}\dfrac{f(x+h)-f(x)}{h}$$ For we evaluate this limit to find a derivative, we say we perform $\textit{Differentiation from First Principles}$.

## The Derivative when $x=a$

The gradient of the tangent to $y=f(x)$ at the point where $x=a$ is denoted $f'(a)$, $$f'(a)=\lim_{h \rightarrow 0}\dfrac{f(a+h)-f(a)}{h}$$

### Example 1

Use the definition of $f'(x)$ to find the derivative of $f(x)=x^2$.

### Example 2

Use the definition of $f'(x)$ to find the derivative of $f(x)=3x^2$.

### Example 3

Use the definition of $f'(x)$ to find the derivative of $f(x)=x^2+1$.

### Example 4

Use the definition of $f'(x)$ to find the derivative of $f(x)=x^2-x$.

### Example 5

Use the definition of $f'(x)$ to find the derivative of $f(x)=x^3$.

### Example 6

Use the definition of $f'(x)$ to find the derivative of $f(x)=\sqrt{x}$.

### Example 7

Use the definition of $f'(x)$ to find the derivative of $f(x)=\dfrac{1}{x}$.

Related YouTube video lesson: Differentiation from First Principles