Lecture 12: The Derivative as a Function

Notations

f1(x)=y1=dydx=dfdx=ddxf(x)=Dxf(x)f^1(x) = y^1 =\frac{dy}{dx} = \frac{df}{dx} =\frac{d}{dx}f(x) = D_xf(x)

to indicate the value of a derivative at a specific number x=ax=a

f1(a)=dydxa=a=dfdxx=a=ddxx=af^1(a)=\frac{dy}{dx} \int_{a=a} = \frac{df}{dx}\int_{x=a} = \frac{d}{dx}\int_{x=a}

Non differentiate functions and their graphs

a)f(x)=x3f(x) = |x-3|  at x=ax=a → A sharp turn or point slope is undefined

b) f(x)=x23f(x)= x^{\frac{2}{3}} at x=x= 0 → There's a "cusp" slope = undefined

c) f(x)=x13f(x) = x^{\frac{1}{3}}  at x=0x=0 → Vertical tangent line slope = ∞ therefore undefined

d) f(x)=x2x2f(x) = \frac{\left | x-2 \right |}{x-2} at x=2x=2 → Non continuous graph is undefined

Theorem - If f is differentiable at x=ax = a  therefore f is continuous at x=ax = a ,but the converse is not always true




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