# Lecture 12: The Derivative as a Function

Latest Version

Published 3 years ago

Latest Version

Published 3 years ago

### Notations

$f^1(x) = y^1 =\frac{dy}{dx} = \frac{df}{dx} =\frac{d}{dx}f(x) = D_xf(x)$

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

$f^1(a)=\frac{dy}{dx} \int_{a=a} = \frac{df}{dx}\int_{x=a} = \frac{d}{dx}\int_{x=a}$

### Non differentiate functions and their graphs

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

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

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

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

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

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