High School Calculus/The Derivative

The definition of a Derivative of a Function

$f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$

Example

$f(x)=x^2$
Use the limit definition with the given function

$f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{(x+\mathrm{\Delta }x{)}^2-x^2}{\mathrm{\Delta }x}$

$f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{(x^2+2x\mathrm{\Delta }x+\mathrm{\Delta }{x}^2)-x^2}{\mathrm{\Delta }x}$

$f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{2x\mathrm{\Delta }x+\mathrm{\Delta }{x}^2}{\mathrm{\Delta }x}$

$f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\mathrm{\Delta }x(2x+\mathrm{\Delta }x)}{\mathrm{\Delta }x}$

$f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{\lim }(2x+\mathrm{\Delta }x)$

$f^{\prime }(x)=2x+0$

$f^{\prime }(x)=2x$