Geometrically

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Geometrically, the value of

\displaystyle\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

, if it exists, gives (a) the slope of the normal to

y=f(x)

(x,\,f(x))

(b) the slope of the secant through

(x, f(x))

and

(x+h,f(x+h))

y=f(x)

(x,\,f(x))

(d) the instantaneous rate of change of

x

with respect to

y

(x,\,f(x))

Related theory

Derivative Foundations
Differentiation from first principles
Intuitive idea of derivatives
Limits and Derivatives

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