Differentiation
Course Material
Gyeongsang National University
Dept. of Information & Communication Engineering
Differentiation
• Differentiation (微分)
– Mathematical technique used to analyze changes in function – Analyze how rapidly a function changes at a particular point
• Voltage drop in capacitors, temperature change of chemical compounds, etc.
Differentiation
• Average rate of change of function for interval
– Average rate of change for interval [t1, t2]
Change of function Change of interval
Differentiation
• Average rate of change of function for interval
– Where should we move t2 to obtain the instantaneous rate of change (slope of tangent) at point A?
Limit of Function
• Limit of Function
Limit of Function
• Example) Derive the limit at each point for 𝑦(𝑥)𝑦 𝑥 = 1 − 𝑥, 𝑥 < 0 3, 𝑥 = 0 𝑥 + 1, 𝑥 > 0 (𝑎) lim 𝑥→3𝑦 (𝑏) lim𝑥→−1𝑦 (𝑐) lim𝑥→0𝑦 If lim
𝑥→𝑎𝑦(𝑥) ≠ 𝑦(𝑎), the function 𝑦(𝑥) is discontinuous at point
Limit of Function
• Example) Answer the question for the following function 𝑦(𝑥).
𝑦 𝑥 = 0, 𝑥 < 0 𝑥, 0 < 𝑥 ≤ 2 𝑥 − 2, 𝑥 > 2 (𝑎) lim 𝑥→2+𝑦 (𝑏) lim𝑥→2−𝑦
At point 𝑥 = 𝑎, the limit does not exist when the left-side limit and the right-side limit are different.
• Limit of Function
– At point 𝑥 = 𝑎, the limit exists when the left-side limit and the right-side limit are equal.
• Continuity of Function
– If lim
𝑥→𝑎 𝑦(𝑥) ≠ 𝑦(𝑎), the function 𝑦(𝑥) is discontinuous at point 𝑥 = 𝑎.
Rate of Change at Point
• Find the average rate of change in 𝑦 𝑥 = 3𝑥
2+ 2
for the following intervals
Rate of Change at Point
• For the interval [3, 3 + 𝛿𝑥], when 𝛿𝑥 approaches to
0, find the rate of change in 𝑦 𝑥 = 3𝑥
2+ 2 at 𝑥 = 3.
Rate of Change at Point
• Rate of Change at Point 𝑥
• Derivative at arbitrary point 𝑥
• Derivative at particular point 𝑥
0Derivative
Existence of Derivative
• The point that the derivative does not exist
– Discontinuous point
– Cusp or corner (point that the instantaneous rates of change from the left-side and right-side are different)
Existence of Derivative
• Example) For the following functions, find the point that
does not exist the derivative
Linearity Rules
• Differentiation is a linear operator
– Differentiation of the sum or difference of functions
– Differentiation of a function multiplied by a constant
Product Rule
• Product rule
– When 𝑦 𝑥 = 𝑢 𝑥 𝑣(𝑥), derivative of 𝑦 𝑥 is given by
Alternative ∶ 𝑑𝑦 𝑑𝑥 = 𝑑𝑢 𝑑𝑥 𝑣 + 𝑢 𝑑𝑣 𝑑𝑥 – Example) a) 𝑦 = 𝑥 sin 𝑥 b) 𝑦 = 𝑡2𝑒𝑡
– Damped sinusoidal signal 𝑓(𝑡) = 𝑒−0.1𝑡 cos 𝑡
Quotient Rule
• Quotient rule
– When 𝑦 𝑥 = 𝑢(𝑥) 𝑣(𝑥) , derivative of 𝑦 𝑥 is given by – Example) a) 𝑦 = sin 𝑥 𝑥 b) 𝑦 = 𝑡2 2𝑡+1 𝑦′ 𝑥 = 𝑣 𝑥 𝑢 ′ 𝑥 − 𝑢 𝑥 𝑣′ 𝑥 𝑣2 𝑥Chain Rule
• Chain rule
– When 𝑦 = 𝑦(𝑧), 𝑧 = 𝑧 𝑥 , 𝑦 can be represented by a function of 𝑥 and 𝑑𝑦 𝑑𝑥 is given by – Example) a) 𝑦 = 𝑧6 , 𝑧 = 𝑥2 + 1 b) 𝑦 = ln(3𝑥2 + 5𝑥 + 7) c) when 𝑦 = ln 𝑓(𝑥), 𝑑𝑦 𝑑𝑥 = 𝑓′(𝑥) 𝑓(𝑥)
𝑑𝑦
𝑑𝑥
=
𝑑𝑦
𝑑𝑧
×
𝑑𝑧
𝑑𝑥
Parametric Differentiation
• Parametric differentiation
– When 𝑦 and 𝑥 are given by the functions of 𝑡 as 𝑦 = 𝑦(𝑡), 𝑥 = 𝑥 𝑡
– We call 𝑡 parameter. Useful when 𝑡 is hard to represented as a function of 𝑥 by removing 𝑡
– Example)
a) When 𝑦 = (1 + 𝑡)2 and 𝑥 = 2𝑡, find 𝑑𝑦
𝑑𝑥
b) When 𝑦 = 𝑒𝑡 + 𝑡 and 𝑥 = 𝑡2 + 1, find 𝑑𝑦
𝑑𝑥
c) When 𝑥 = sin 𝑡 + cos 𝑡 and 𝑦 = 𝑡2 − 𝑡 + 1, find 𝑑𝑦
𝑑𝑥 𝑡 = 0
𝑑𝑦
𝑑𝑥
=
𝑑𝑦
𝑑𝑡
×
𝑑𝑡
𝑑𝑥
=
𝑑𝑦
𝑑𝑡
/
𝑑𝑥
𝑑𝑡
Differentiation of Implicit Form
• Differentiation of implicit Form
– When 𝑥3 + 𝑦 = 1 + 𝑦3, what is 𝑑𝑦 𝑑𝑥?
– Hard to represent as 𝑦 = 𝑓 𝑥
𝑦 is represented in an implicit form of 𝑥 – Differentiation of both sides with 𝑥
– Example)
a) When 𝑥3 + 𝑦 = 1 + 𝑦3, find 𝑑𝑦
𝑑𝑥
b) When ln 𝑦 = 𝑦 − 𝑥2, find 𝑑𝑦
Logarithmic Differentiation
• Logarithmic Differentiation
– Differential on products of complex functions can be easily obtained using logarithmic differentiation
– Differentiation by applying the natural logs (ln) – Example)
a) When 𝑦 = 𝑥3(1 + 𝑥)9𝑒6𝑥, find 𝑑𝑦
𝑑𝑥?
b) When 𝑦 = 1 + 𝑡2sin2𝑡, find 𝑑𝑦
Higher Order Derivatives
• Higher order derivatives
– We call 𝑑𝑦
𝑑𝑥 as the first derivative of 𝑦(𝑥)
– We call 𝑑
𝑑𝑥 𝑑𝑦 𝑑𝑥 =
𝑑2𝑦
𝑑𝑥2 as the second derivative of 𝑦(𝑥) (can be
represented by 𝑦′′) – We call 𝑑𝑛𝑦
𝑑𝑥𝑛 as 𝑛-th order derivative of of 𝑦(𝑥)
– Example)
a) When 𝑦 𝑥 = 3𝑥2 + 8𝑥 + 9, find 𝑦′ and 𝑦′′ b) When 𝑦 𝑡 = 2 sin 3𝑡, find 𝑦′ and 𝑦′′
Higher Order Derivatives
• Shape of the function according to the first and second
derivatives
(a) y is convex up (y > 0, y > 0); (b) y is concave up (y > 0, y < 0); (c) y is concave down (y < 0, y < 0); (d) y is convex up (y < 0, y > 0).
Local Maximum and Local Minimum
• Local maximum and local minimum
– Local maximum – the point at which the function stops to increase and begins to decrease.
– Local minimum – the point at which the function stops to decrease and begins to increase.