The Derivative

Definition

Example (1) Find the derivative of f(x) = 4 at any point x

Example (2) Find the derivative of f(x) = 4x at any point x

Example (3) Find the derivative of f(x) = x 2 at any point x

Example (4) Find the derivative of f(x) = x 3 at any point x

Example (5) Find the derivative of f(x) = x 4 at any point x

Example (6) Find the derivative of f(x) = 3x 3 + 5x 2 - 2x + 7 at any point

Questions Find from the definition the derivative of each of the following functions: 1. f(x) = √x 2. f(x) = 1/x 3. f(x) = 1/x 2 4. f(x) = 5 / (2x + 3)

Power Rule Let: f(x) = x n, where n is a real number other than zero Then: f'(x ) = n x n-1 If f(x) = constant, then f ' (x) = 0

Algebra of Derivatives

Example (1)

Solution

The Chain Rule The derivative of composite function for the case f(x) = g n (x) Let: f(x) = g n (x) Then: f ' (x) = ng n-1 (x). g ' (x) Example: Let f(x) = (3x 8 - 5x + 3 ) 20 Then f(x) = 20 (3x 8 - 5x + 3 ) 19 (24x 7 - 5)

Examples (1)

Example (2)

Solution

Example (3)

Homework

Answers of Questions (1) Find from the definition the derivative of each of the following functions: 1. f(x) = √x 2. f(x) = 1/x 3. f(x) = 1/x 2 4. f(x) = 5 / (2x + 3)

1

2

3

4.

Differentiability & Continuity 1. If a function is differentiable at a point, then it is continuous at that point. Thus if a function is not differentiable at a point, then it cannot be continuous at that point. But the converse is not true. A function can be continuous at a point without being differentiable at that point. 2. A point of the graph at which the graph of the function has a vertical tangent or a sharp corner is a point where the function is not differentiable regardless of continuity

Examples (1) Sharp Corner This function (Graph it!) is continuous at the point x=2, since the limit and value of the function at that point are equal ( Show that!) but it is not differentiable at that point, since the right derivative of f at x=2 is not equal to the left derivative a that point.

Examples (2) Vertical Tangent When both right and left derivatives are +∞ or both are - ∞ This function (Graph it!) is continuous at the point x=0, since the limit and value of the function at that point are equal ( Show that!) but it is not differentiable at that point, since the right derivative (and also the left derivatives) of f do not exist at x=2 ( both are +∞)

Examples (3) Cusp When the one of the one-sided derivative is +∞ and the other is- ∞ This function (Graph it!) is continuous at the point x=0, since the limit and value of the function at that point are equal ( Show that!) but it is not differentiable at that point, since at x=2 the right derivative does not exist ( is + ∞) and also the left derivatives does not exist and is -∞)

Limits Involving Trigonometric Functions

All trigonometric functions are continuous a each point of their domains, which is R for the sine & cosine functions(→The limit of sinx and cosx at any real number a are sina and cosa respectively ), R-{ π/2, - π/2, 3π/2, - 3π/2,…………} for the Tangent and the Secant functions and R-{0, π, - π, 3π, - 3π,…………} for the Cotangent and the Cosecant functions.

Important Identity

Examples (1)

Example (2) f is continuous for all x other than zero. To check, whether it is continues, as well at x=0, we need to that its limit at x=0 is equal to f(0), which is given as zero. Solution

Example (3) For f to be continuous at x=0, we need its limit at x= 0 to exist and to equal the value at that point, which is 9. Since its right limit at x=0 is equal also 9, it remains that its left limit at that point be equal to that value. Solution

Example (4) All members are continuous for all x other than 2. For a member to be continuous at x=2, we need the limit of the member function at x= 2 to exist and to equal the value at that point, which is 1/2c. Since the right limit at x=2 of any member is equal o 1/2c, it remains that its left limit at that point be equal to that value. Solution