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Differential Calculus

Chapter 1 · From Limits to Derivatives

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Derivative — first principles
f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
The limit of the secant gradient as the gap h→0h \to 0.
Geometric meaning
f′(x)f'(x) = slope of the tangent to y=f(x)y=f(x) at that point = instantaneous rate of change.
Notation
f′(x)  =  dydx  =  y˙f'(x) \;=\; \frac{dy}{dx} \;=\; \dot{y}
All three mean the same derivative.
Worked results (from first principles)
x2→2x,x3→3x2,1x→−1x2x^2 \to 2x, \quad x^3 \to 3x^2, \quad \tfrac{1}{x} \to -\tfrac{1}{x^2}
Derivative (first principles)
f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h\to0}\tfrac{f(x+h)-f(x)}{h}
Meaning
Gradient of the tangent / instantaneous rate of change.
Power rule
ddxxn=nxn−1\tfrac{d}{dx}x^n = n x^{n-1}
Common trap
Differentiable ⇒ continuous, but not vice-versa (corners/cusps).