High School Calculus/The Derivative

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The definition of a Derivative of a Function

f'(x)=\lim_{{\Delta}x\rightarrow 0} \frac{f(x+{\Delta}x)-f(x)}{{\Delta}x}

Example

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

f'(x)=\lim_{{\Delta}x\rightarrow 0} \frac{(x+{\Delta}x)^2-x^2}{{\Delta}x}

f'(x)=\lim_{{\Delta}x\rightarrow 0} \frac{(x^2+2x{\Delta}x+{\Delta}x^2)-x^2}{{\Delta}x}

f'(x)=\lim_{{\Delta}x\rightarrow 0} \frac{2x{\Delta}x+{\Delta}x^2}{{\Delta}x}

f'(x)=\lim_{{\Delta}x\rightarrow 0} \frac{{\Delta}x(2x+{\Delta}x)}{{\Delta}x}

f'(x)=\lim_{{\Delta}x\rightarrow 0} (2x+{\Delta}x)

f'(x)=2x+0

f'(x)=2x