Page 15 - mathematics

* Definition of Derivative: Important theorems:

The first derivative of the function y  f  x  with 1. If y = f(x) is differentiable at x = a, then y = f(x) is
respect to the variable x is the function f  whose value continuous at a . The inverse is not always true.

at x is: 2. If the function y = f(x) is discontinuous at the point
x = a, then it is not differentiable at this point.
f x   y   dy  lim f x hf x 
dx h 0
h

provided the limit exists.

Geometric Interpretation of Derivative: * we can write the equation of the tangent line to the
curve at the point (a, f(a)):
* The slope of tangent line to the graph of the function
f(x) at (a,f(a)) is the derivative of f(x) at x = a. y  f a  f a x  a
y
y  f x  Example

. P  f ah f a Find an equation of the tangent line to the curve
h
y  x2  8x  9 at the point (3,- 6).

Solution

a ah x y  2x  8 y 3  23  8  2

f a   lim f a  h  f a y  2x
h 0
h y   6  2 x  3

= Slope of tangent at P

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