First Principles

The First Principles defines how basic rule of derivative of a function.
$$\displaystyle f'(x) = \lim_{h\to\infty} \frac{f(x+h)-f(x)}{h}$$

Patterns of the First Principles

Pattern 1

Differentiate $f(x) = x^2$ from the first principles.

Pattern 2

Differentiate $f(x) = x^3$ from the first principles.

Pattern 3

Differentiate $f(x) = \sqrt{x}$ from the first principles.

Pattern 4

Differentiate $\displaystyle f(x) = \frac{1}{x}$ from the first principles.

Pattern 5

Differentiate $\displaystyle f(x) = \sqrt[3]{x}$ from the first principles.