### Introduction to Derivatives

The derivative of a function

‘f(x)’with respect to variable‘x’is the functionf ‘(x)whose value atxis

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

Previous page:

-Limits of a Function

http://www.electricalengineering4u.com/applied-calculus/limits-of-a-function/