Calculus
Vimukthi Pathirana
MEng. (AIT), BSc (Hons) Eng. (AIT), reading MBA(UOP)
Introduction to Differentiation
• Calculus – Study of CHANGE
• Differential Calculus (or Differentiation)
• Process of finding the derivative, or rate of change, of a function
Introduction to Differentiation
• Tangent to a Curve
• A tangent line to a curve at a point is that
straight line that best approximates the curve
near that point.
• A line or plane that touches a curve or a cured
surface at exactly one point
Introduction to Differentiation
• Gradienty of a Curve
• If a tangent is drawn at a point P on a curve,
the gradient (slope) of this tangent is said to
be the gradient of the curve at P.
• The gradient of the curve at P is equal to the
gradient of the tangent PQ
Introduction to Differentiation
• Gradienty of a Curve
• For the curve show in figure, let the point
A and B have coordinates 𝑥1 , 𝑦1 and
𝑥2 , 𝑦2 , respectively.
• In function notation,
𝑦1 = 𝑓(𝑥1 )
𝑦2 = 𝑓(𝑥2 )
Introduction to Differentiation
• Gradienty of a chord AB
𝐵𝐶 𝐵𝐷 − 𝐶𝐷 𝑓 𝑥2 − 𝑓(𝑥1 )
= = =
𝐴𝐶 𝐸𝐷 𝑥2 − 𝑥1
Introduction to Differentiation
• Example: for the curve 𝑓 𝑥 = 𝑥 2
(a) The gradient of the chord AB
𝑓 3 −𝑓(1) 9−1
= = =4
3−1 2
(b) The gradient of the chord AC
𝑓 2 −𝑓(1) 4−1
= = =3
2−1 1
Introduction to Differentiation
(a) The gradient of the chord AD
𝑓 1.5 −𝑓(1) 2.25−1
= = = 2.5
1.5−1 0.5
(b) The gradient of the chord AE
𝑓 1.1 −𝑓(1) 1.21−1
= = = 2.1
1.1−1 0.1
Introduction to Differentiation
• As Point B move closer and closer
to point A, the gradient of the chord
approaches nearer and nearer to
the value 2.
• This is called the limiting value of
the gradient of the chord AB, when
B coincides with A, there the chord
becomes the tangent to the curve
Methods of Differentiation
• Differentiation from the first
principle
• Assuming A and B are two very close
points on a curve,
• δx (delta x) and δy (delta y)
representing small increments in the
x and y directions
Methods of Differentiation
• Differentiation from the first
principle
𝛿𝑥
• Gradient chord 𝐴𝐵 =
𝛿𝑦
𝛿𝑦 = 𝑓 𝑥 + 𝛿𝑥 − 𝑓 𝑥
𝛿𝑦 𝑓 𝑥+𝛿𝑥 −𝑓 𝑥
=
𝛿𝑥 𝛿𝑥
Methods of Differentiation
• Differentiation from the first
principle
𝛿𝑦 𝑓 𝑥+𝛿𝑥 −𝑓 𝑥
=
𝛿𝑥 𝛿𝑥
• As 𝛿𝑥 approaches to zero, gradient
of the chord approaches to the
gradient of the tangent at A
Methods of Differentiation
• Differentiation from the first principle
𝛿𝑦 𝑓 𝑥 + 𝛿𝑥 − 𝑓 𝑥
lim = lim { }
𝛿𝑥→0 𝛿𝑥 𝛿𝑥→0 𝛿𝑥
𝑑𝑦 𝛿𝑦
In Leibniz notation, = lim
𝑑𝑥 𝛿𝑥→0 𝛿𝑥
𝑓 𝑥+𝛿𝑥 −𝑓 𝑥
In Function notation, 𝑓 ′ (𝑥) = lim { }
𝛿𝑥→0 𝛿𝑥
Methods of Differentiation
• Differentiation from the first principle
𝑑𝑦
is the same as 𝑓 ′ (𝑥) and is called the differential coefficient or the
𝑑𝑥
derivative.
The process of finding differential coefficient is called differentiation
𝑑𝑦 ′
𝛿𝑦 𝑓 𝑥 + 𝛿𝑥 − 𝑓 𝑥
= 𝑓 (𝑥) = lim = lim { }
𝑑𝑥 𝛿𝑥→0 𝛿𝑥 𝛿𝑥→0 𝛿𝑥
Methods of Differentiation
• Example: differentiate 𝑓 𝑥 = 𝑥 2 from the first principle
𝑓 𝑥 = 𝑥2
𝑓 𝑥 + 𝛿𝑥 = (𝑥 + 𝛿𝑥)2
𝑓 𝑥+𝛿𝑥 −𝑓 𝑥 (𝑥+𝛿𝑥) 2 −𝑥 2
𝑓 ′ (𝑥) = lim { } = lim { }
𝛿𝑥→0 𝛿𝑥 𝛿𝑥→0 𝛿𝑥
′ 𝑥 2 +2𝑥𝛿𝑥+(𝛿𝑥)2 −𝑥 2 2𝑥𝛿𝑥+(𝛿𝑥)2
𝑓 (𝑥) = lim { } = lim { }
𝛿𝑥→0 𝛿𝑥 𝛿𝑥→0 𝛿𝑥
= lim 2𝑥 + 𝛿𝑥
𝛿𝑥→0
Methods of Differentiation
• Example: differentiate 𝑓 𝑥 = 𝑥 2 from the first principle
𝑓 ′ (𝑥) = lim 2𝑥 + 𝛿𝑥
𝛿𝑥→0
as 𝛿𝑥 → 0, {2𝑥 + 𝛿𝑥} → {2x + 0}
Thus, 𝑓 ′ (𝑥) = 2𝑥
∴ the differential coefficient of 𝑥 2 is 2𝑥.
Methods of Differentiation
• Differentiation of 𝑦 = 𝑎𝑥 𝑛 by the general rule
From differentiation by first principles, a general rule for differentiating
𝑎𝑥 𝑛 emerges where a and n are any constants.
This tule is,
Methods of Differentiation
• Differentiation of 𝑦 = 𝑎𝑥 𝑛 by the general rule
When differentiating, results can be
expressed in a number of ways;
Methods of Differentiation
• Differentiation of 𝑦 = 𝑎𝑥 𝑛 by the general rule
some laws if indices you need
to know;
Methods of Differentiation
• Exercises:
1. Differentiate with respect to x: y = 3x 8
2
2. Differentiate with respect to x: y =
x3
3. Differentiate with respect to x: y = 6 x
4. Differentiate, y = 7
5. Differentiate, y = 11x
1 5 4
6. Find the differential coefficient of, y = x − 2 + 14
3 x
3 ′ (𝑡)
7. If, f t = t + 3
find 𝑓
t
dy 5x2 −2x
8. Determine given, y =
𝑑𝑥 3x
Methods of Differentiation
• Exercises:
3 2 3
9. Find the differential coefficient of, y = x − 4 + 4 x6 + 2
5 x
(x+1)3
10. Differentiate y = , with respect to x
3x
5
11. Find the gradient of the curve 𝑦 = 2𝑥 3 − at x = 2
𝑥
12. Find the gradient of the curve 𝑦 = 2𝑥 5 − 3𝑥 2 + 5𝑥 − 4 at the
points (0, -2) and (1, 4)
Derivative Formulas
