What is differentiation?
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Differentiation refers to the process of finding a derivative. A function that has a derivative at a point is said to be differentiable at that point.
So what is a derivative? A
derivative is the slope of a particular curve at a specified point.
This is also known as:
- > The slope of the tangent line
- > Instantaneous rate of change
If the point of tangency is (a, f(a)), the first derivative is denoted by f'(a).
Definition of the derivative:
f'(x) = lim f(x + h) - f(x)
h→0 h
> If f'(x) exists → f is differentiable at x!
Alternate form of the derivative:
f'(a) = lim f(x) - f(a)
x→a x-a
> For all a for which the limit exists, f'(x) is a function of a.
Okay, but how do we know if a derivative exists at a point?
f'(a) exists at the point if the left and right limits of the slope are both equal to the same finite value at that point.
f'(a) fails to exist at the point if there is a corner, a cusp, a vertical tangent line, or a discontinuity.
* Differentiability implies continuity!