Derivative from first principles
Objective. Derive differentiation rules from difference quotients.
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- 2. Worked example
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Learn: the key idea
The derivative of f at x is defined as the limit of the difference quotient f'(x)=lim(h→0) [f(x+h)−f(x)]/h, representing the instantaneous rate of change or slope of the tangent line. Standard rules like the power rule are themselves derived from this limit rather than assumed. For f(x)=x², the difference quotient is [(x+h)²−x²]/h = (2xh+h²)/h = 2x+h, and letting h→0 gives f'(x)=2x. This first-principles approach confirms shortcut rules and is required whenever a function's differentiability must be justified rigorously, such as at a corner or kink. Always simplify the quotient algebraically before taking the limit, since direct substitution gives 0/0.
Worked example
Find f'(x) for f(x)=x²+3x using the derivative from first principles.
- 1Write the difference quotient: [f(x+h)−f(x)]/h = [(x+h)²+3(x+h)−x²−3x]/h.
- 2Expand the numerator: x²+2xh+h²+3x+3h−x²−3x = 2xh+h²+3h.
- 3Divide by h: (2xh+h²+3h)/h = 2x+h+3.
- 4Let h→0 to get f'(x)=2x+3.
Answer: f'(x) = 2x + 3
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Start practisingCommon mistake. Students substitute h=0 into the difference quotient before simplifying algebraically, getting the meaningless 0/0 and abandoning the method, instead of expanding and cancelling h first.
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