# Lecture 12: The Derivative as a Function

### 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