3.0 Differentiability

  Continuity and Differentiability

    3.0 Differentiability

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3.0 DifferentiabilityA function $y=f(x)$ is said to be differentiable at any point $x=c$ if the derivative of the function exists at that point i.e., $f'(x)$ exists at every point in its domain which is written as $${\left( {\frac{{dy}}{{dx}}} \right)_{x = c}} = f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(h)}}{h}$$So, we can say that for a function to be differentiable at any point in its domain, right hand derivative $$\mathop {\lim }\limits_{h \to {0^ + }} \frac{{f(x + h) - f(h)}}{h}$$ and left hand derivative $$\mathop {\lim }\limits_{h \to {0^ - }} \frac{{f(x + h) - f(h)}}{h}$$Both must be finite and equal.