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