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.
Find by definition the derivative of for x>0
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)
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.
-Limits of a Function