← MATH 01130 · Calculus I
MATH 01130 · Derivatives
Definition as limit of difference quotient
Definition
f'(a) = lim h→0 [f(a+h)−f(a)]/h, when the limit exists. Geometrically: slope of the tangent line; physically: instantaneous rate of change.
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The difference quotient is the slope of a secant. Sending h→0 zooms the secant into a tangent.
Alternate form: lim x→a [f(x)−f(a)]/(x−a). Same idea.
If the limit DNE (corner, cusp, jump, vertical tangent), f is not differentiable at a. Differentiability ⇒ continuity, but not conversely.
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1.Use the limit definition to find f'(2) for f(x)=x².