Embed Size (px)
Citation preview
1. The concept of derivative - Notation - First Principle of Differentiation2. Rules of Differentiation for: - Derivative of a Constant - Derivative of xn - Constant Factor Rule - Derivative of a Sum or Differences - Product & Quotient Rules - Chain Rule and Power Rule - Exponent & Logarithmic Rules3. Higher order of derivatives• Critical points – minimum, maximum, inflection point• Application : Business and economics
After finishing this class, you should be able to:
• Explain the concept of derivative.• Differentiate a function using the First Principle
(The Concept of limit)
The concepts of derivative : NotationIf f defined as the function of x and can be written as f(x). Then the derivative of f(x) = y denoted as f’(x) or is read as
“derivative value of function f at x”.
The process to get f’(x) is called DIFFERENTIATION (FIRST DERIVATIVE)
dx
dy
f(x)
g(u)
y = f(x)
U = f(v)
f’(x)differentiate
g’(u)differentiate
dydx
differentiate
dUdv
differentiate
y/f(x)
x3 6
15
30
0
Slope of a straight line
The slope for the line, m ism = 30 – 15 6 – 3 m = 5
m=5
A
m=5
B
m=?
C
The slope (m) of a straight line is always consistent at any points on the line.
Slope of a curve• A curve is not like a straight line – it does not have a consistent slope.• Slope for a curve can be obtained by drawing a tangent line at any point of
measurement on the curve.• The slope of the tangent line is used to represent the slope of a curve at the
point it is drwan.• Therefore, the slope for a curve vary accordingly to the point where it is
measured.
0
y/f(x)
x
Tangent
The slope of a curve (at a certain point on that curve) can be obtain by measuringthe slope of tangent line at that point.
m = 1 2
1
2
a
b(a,b)
The slope for function f at the point(a,b) is ½.
Slope of a curve
The slope of the function f at the point (c,d) is 1.
1
1
m = 1 1
c
d(c,d)
The slope of a curve (at a certain point on that curve) can be obtain by measuringthe slope of tangent line at that point.
Slope of a curve
The slope for function f at the point(e,f) is ? 6
2
m = ?
e
f(e,f)
The slope of a curve (at a certain point on that curve) can be obtain by measuringthe slope of tangent line at that point.
Slope of a curve
x
h
x+h
f(x)
f(x+h)
Consider a function, f and suppose that there are 2 points (A and B) on the function (curve).
A
B
= (x, f(x))
= (x+h, f(x+h))
x
h
x+h
f(x)
f(x+h)
A= (x, f(x))
B= (x+h, f(x+h))
The tangent touched the curve at only one point (A)
A secant line touched the curve at 2 points (A and B)
From the diagram:- PQ is the tangent for the function f at the point A (green line)- AB is a secant line that touched the function f at the point A and B (grey line)
P
Q
P
Q
The slope for AB chord , (mab) is:
f(x)
f(x+h)
A= (x, f(x))
B= (x+h, f(x+h))
y2 – y1 = f(x+h) – f(x) or f(x+h) – f(x)x2 –x1 (x+h) – (x) h
x x+h
The slope for PQ tangent is an approximation of chord AB to the tangent, that is when h is approaching 0.
P
Q
[email protected], we can see that the slope for PQ tangent (mpq)is derive from: = lim Slope for AB OR lim f(x+h) – f(x) h0 h0 h
x
h
x+h
f(x)
f(x+h)
A
B
= (x, f(x))
= (x+h, f(x+h))
P
Q
Thus, the slope for function f at the point A is EQUAL to
The slope for PQ tangent that is lim f(x+h) - f(x), therefore h0 h
x
h
x+h
f(x)
f(x+h)
A
B
= (x, f(x))
= (x+h, f(x+h))
P
Q
We called lim f(x+h) – f(x) as Differentiation Using The First Principle h0 h And is denoted by f’(x) OR dy/dx [email protected]
