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IB Maths .Basic differentiation
Calculate the following limits:
By the end of the lesson you will be able to:
• Understand the definition of derivative.
• Find derivatives from first principles.
Differentiation
• Calculate the gradient of a curve without drawing 
the tangent line.
• Relate derivatives and slopes of curves.
Introduction to Differentiation
http://www.youtube.com/watch?v=QC5ITOflh3k&feature=related
Gradient of a line
B
A
Gradient of a line
B
A
y2
y1
x1 x2
Gradient of a line
B
A
y2
y1
x1 x2
The gradient of a line is constant.
Gradient of a curve
The gradient of a curve at a point  is the slope of 
the tangent line at that point.
http://www.math.umn.edu/%7Egarrett/qy/TraceTangent.html
Tangent line to a curve.ggb
Gradient of a curve
The gradient of a curve
varies at each point.
We need to find a method to calculate the slope of 
the tangent line at any point.
x
Let's consider a point P on the curve.
P
f(x)
Coordinates of P :      ( x ,  f(x) )
f(x)
Secant to tangent.ggb
Coordinates of Q :      ( x+h ,  f(x+h) )
x
Let's consider another point on the curve, Q.
P
f(x)
Q
x + h
f(x)
Coordinates of Q :      ( x+h ,  f(x+h) )
x
Let's take another point on the curve , Q.
P
Q
x + h
f(x+h)
f(x)
x
The line PQ is  secant  to the curve. If we find the 
gradient of this line , it is not the tangent line but is a 
starter.
P
f(x)Q
x + h
x
P
f(x)Q
x + h
f(x)
f(x+h)
m =
x
P
f(x)Q
x + h
f(x)
f(x+h)
Gradient of secant line PQ :
We can rewrite this gradient in a different way :
Gradient of secant PQ :
Q
x + h
Veamos que sucede si Q se acerca a P...
P
x
Secant to tangent.ggb
slope of the secant =
x
P
f(x)Q
x + h
If Q gets closer to P... 
Then we need h to be as small as possible.
h 0When the slope of the secant tends to
be the slope of the tangent.
This is the difference quotient , the definition of 
the derivative.
This  is called
the  first derivative of function f with respect to x.
Notation:
f '(x)    or   y'     or 
Find the derivative of the function      f(x) = x2
Pp
R
S
We will find the gradient of the tangent at any 
point for the function  f(x) = x2
f(x)  = x 2
Pp
R
S
m =4
m =-6
We found for f(x) = x2
which is a new function that gives the
value of the gradient of the curve at each point.
Para
its derivative:
Calculate the derivative of  f(x) = 3 x2
 +1
To practice this topic on-line :
http://archives.math.utk.edu/visual.calculus/2/definition.7/index.html
The process of finding the derivative is called 
differentiation.
Book page 357 Ex 12B
http://www.calculus­help.com/funstuff/phobe.html
At the end of the lesson:
 
http://www.calculus­help.com/funstuff/phobe.html
To revise this lesson at home:
At the end of the lesson:
http://animoto.com/play/hVIw4sOG1tEL3hYONxJyJQ
Attachments
Tangent line to a curve.ggb
Secant to tangent.ggb

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