Introduction to Derivatives

The derivative of a function ‘f(x)’ with respect to variable ‘x’ is the function f ‘(x) whose value at x is

Provided the limits exist

If f ‘(x) exists we say that f has derivative at ‘x’ . The process of calculating derivative is called differentiation.

Example

Find by definition the derivative of    for x>0

Notations

Newton:    f ‘ ,  y’

Leibnitz:   Dy , d/dx

Note: Every differential function has continuity but every continuity don’t have differentiability.

Rules of Differentiation

1- d/dx (c) = 0        Where ‘c’ is any constant.

2- y = x^n  it’s derivative is dy/dx = n x^n-1  (Power Rule)

y = (ax+b)^n it’s derivative is dy/dx = n(ax+b)^n-1 . a  (Extended power rule)

3- y = c f(x) it’s derivative is dy/dx = c f ‘(x)

4- y = u+v it’s derivative is dy/dx = du/dx + dv/dx    Where u and v are functions of x.

5- y = u.v it’s derivative is dy/dx = u.dv/dx + v.du/dx  (Product Rule)

6- y = u/v it’s derivative is dy/dx = (v.du/dx – u.dv/dx) / v^2  (Quotient Rule)

 Examples

Geometrical Meanings of Derivative of a Function

Geometrically derivative of a function at a point is the slope of tangent to the graph of that function at that point.

Standard Derivatives

 

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-Limits of a Function

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