Basic Derivative Results

If y = f(x), then

1)  \frac{{dy}}{{dx}}  = f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}

2) f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(a  + h) - f(a)}}{h} = \mathop {\lim }\limits_{x \to a} \frac{{f(x) - f(a)}}{{x -  a}}

3) L.H.D = f'(a - 0) = \mathop {\lim }\limits_{h \to 0  - 0} \frac{{f(a + h) - f(a)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{f(a  - h) - f(a)}}{{ - h}}

4) R.H.D = f'(a + 0) = \mathop {\lim }\limits_{h \to 0  + 0} \frac{{f(a + h) - f(a)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{f(a  + h) - f(a)}}{h}