Page 12 - (1-1)eng

(1 - 1) The First Derivative of a Function

Differentiation

Definition y =ƒ ( x )

The function ƒ ( x ) is said to be differentiable at x = a if ƒ (a + h)
 

' ( a ) Lim   a h    a exists.

h 0 h ƒ (a)
 h
a a+ h

This limit is called the derivative of  at x = a and

 dy 

denoted by or   y =ƒ ( x )
 dx  x  a ƒ ( x )

Alternative Formula for Derivative

    x   a ƒ (a)
ƒ' ( a ) = Lim .
x  a x a 
a x

The following are all interpretations for any of the perivous limits Slope of the tangent = tan 

1) The slope of the tangent to the curve y = ƒ ( x ) at x = a.
2) The rate of change of the function y = ƒ ( x ) with respect to x at x = a.

3) The derivative of the function y = ƒ ( x ) at x = a.

Left-hand and Right-hand Derivatives (One-sided derivatives)
 a h      a  a h      a
–
+
' ( a ) = Lim , ' ( a ) = Lim
h 0  h h 0  h

+
–
' ( a ) exists  ' ( a ) and ' ( a ) both exist and are equal.
Remember

 If ƒ has derivative at x = a , then ƒ is continuous at x = a

 If ƒ is not continuous at x = a, then it is not differentiable at x = a

 A function may be continuous at a point but not differentiable at that point, for example:

the function ƒ ( x ) = | x | is continuous and not differentiable at x = 0

Calculus 11 Unit (1)