Contents
- 1 Who Developed Modern Calculus in the 17th Century?
- 1.1 Application of Calculus In Physics
- 1.2 Differentiation
- 1.3 Derivatives of algebraic functions:
- 1.4 Derivatives of trigonometric functions:
- 1.5 Derivatives of Inverse trigonometric or cyclometric functions:
- 1.6 Derivatives of logarithmic and exponential functions:
- 1.7 Basic properties of differentiation:
From the study of subatomic particles to the laws of motion, Physics Topics offer insights into the workings of the world around us.
Who Developed Modern Calculus in the 17th Century?
Application of Calculus In Physics
Calculus is a very important branch of mathematics. In this branch, the main pillar is the infinitesimal magnitudes and multitude of infinitesimal numbers. There is no better tool in mathematics than calculus to express any physical quantity [which is a quantitative property] in mathematical terms. Modern calculus was developed in the 17th century by Issac Newton and Gottfried Wilhelm Leibniz independently. Calculus is a Latin word; it means ‘small pebble used in an abacus for counting’. The word calculus is also used in Latin as a synonym of counting.
In physics, it is important to know the relation among the variable quantities or how the change in one quantity affects another. There is no other way to analyse without the use of infinitesimal magnitudes and numbers. So, in physics, calculus is an indispensible tool.
Variable and constant: A variable is a value that may change within the scope of the given problem or set of operations. A constant is a value that remains unchanged. Suppose, a green grocer has a stock of 10 kg bitter gourd and he sells it at a price of Rs. 16 per kg. If the seller does not change the price, it is constant. But the quantity of bitter gourd bought by individual buyers and its price are variables, because these may vary from 0 kg to 10 kg and from Rs. 0 to Rs. 160 .
Real variable and complex variable: A variable to which only real numbers are assigned as values is called real variable. A variable which can take on the value of a complex number is called complex variable. Any complex variable has two parts—real part and imaginary part. Suppose z(= x + iy) is a complex variable. For this variable, x and iy are the real and imaginary parts respectively. Here, x and y are real variables and i = \(\sqrt{-1}\) is unit imaginary number or the imaginary unit.
We will mainly be using real variables in our discussion.
Function • Independent variable and dependent variable: Any function relates two variables or variable quantities. Of these, one is a dependent variable and the other, an independent variable. Suppose a relation is expressed as y = f(x). We read the equation as: y is a function of x. Here y and x are dependent and independent variables respectively. We generally express any functional relation as:
y = f(x) = ax2 + bx+ c
or, y(x) = ax2 + bx+ c
or, y = ax2 + bx+ c [generally ‘ (x) ’ is not written].
Here, if a, b and c are constants, then for any value of x, we can calculate the corresponding value of y.
Function: If we get only one value of a dependent variable y for a single value of independent variable x, then we can say y is a function of x. Calculus is based on such functions, y = x is a functional relation. But y2 = x is not a functional relation. Actually, y2 = x consists of two functions -y = \(\sqrt{x}\) and y = –\(\sqrt{x}\).
Differentiation
Suppose, y = f(x) is a functional relation, where x and y are respectively the independent and dependent variables. If x increases to x + Δx i.e., if the increment of x is Δx, then y changes to y + Δy i.e., the increment of y is Δy [Fig.].
So we can write it as an equation:
Δy = f(x + Δx) – f(x)
Dividing both sides by Δx, we get
\(\frac{\Delta y}{\Delta x}\) = \(\frac{f(x+\Delta x)-f(x)}{\Delta x}\) ……. (1)
= the change of y caused by a unit change of x.
Now, if Δx → 0, (i.e., the value of Δx tends towards zero or the value x is very small) we can write \(\frac{\Delta y}{\Delta x}\) as
\(\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}\) = \(\frac{d y}{d x}\) = f'(x)
[we read \(\lim _{\Delta x \rightarrow 0}\) as : limit Δx tends to zero]
If Δx → 0, then the limiting value of \(\frac{\Delta y}{\Delta x}\) is expressed as \(\frac{d y}{d x}\) or f'(x). \(\frac{d y}{d x}\) or f'(x) is “the derivative of y with respect to x ”. Actually, \(\frac{d y}{d x}\) is the rate of change of y with respect to x.
The process of determining the derivative is called differentiation. It must be remembered that \(\frac{d y}{d x}\) does not mean dividing dy by dx. It is only the symbol of the limiting process which is shown in equation (2).
It is to be mentioned that, when Δx → 0, the straight line A’B’ is the tangent to the curve y = f(x) at a point A and θ = θ’. Besides \(\frac{d y}{d x}\) or f'(x), we can also express the derivative of y with respect to x with the symbols — y’ or y1 or Dy or \(\frac{d}{d x}\)(y) or \(\frac{d}{d x}\){f(x)}
The meaning of Δx → 0 : The value of Δx tending to 0 means that the value of Δx is never exactly 0. Whatever value close to zero we may imagine, the value of Δx will be even closer to zero. Suppose we imagine a value 0.00001 (or -0.00001) which is nearly 0. In that case, Δx can assume any value between 0.00001 to 0 (or -0.00001 to 0).
Slope: If we consider two points A (x1, y1) and B (x2, y2) on a straight line (line number 1) [Fig.] on a plane xy, then the slope of the straight line
m = \(\frac{y_2-y_1}{x_2-x_1}\) = tanθ
Now, instead of a straight line if we consider a curve (line number 2) then the slope is not equal at all the points on the curve. To measure the slope we need to take two points within a very small distance. The curve between these two points is considered to be a part of a straight line. If the coordinates of these two points P and Q are (x, y) and (x+ dx, y+ dy) respectively, then the slope of the curve at the point (x, y) is
m = \(\frac{(y+d y)-y}{(x+d x)-x}\) = \(\frac{d y}{d x}\)
So we can say, the slope of the curve on a plane xy at a point (x, y) = \(\frac{d y}{d x}\)
Derivatives of algebraic functions:
- \(\frac{d}{d x}\)(xn) = nxn – 1
- \(\frac{d}{d x}\)(axn) = anxn – 1
- \(\frac{d}{d x}\)(ax) = axlna [we can write logea as lna]
Derivatives of trigonometric functions:
- \(\frac{d}{d x}\)(sin x) = cos x
- \(\frac{d}{d x}\)(cos x) = -sin x
- \(\frac{d}{d x}\)(tan x) = sec2x
- \(\frac{d}{d x}\)(cot x) = -cosec2x
- \(\frac{d}{d x}\)(sec x) = secx tanx
- \(\frac{d}{d x}\)(cosec x) = -cosecx cotx
- \(\frac{d}{d x}\)(sinax) = acosax
- \(\frac{d}{d x}\)(cosax) = -asinax
Derivatives of Inverse trigonometric or cyclometric functions:
Derivatives of logarithmic and exponential functions:
- \(\frac{d}{d x}\)(logex) = \(\frac{1}{x}\)
- \(\frac{d}{d x}\)(logax) = \(\frac{1}{x}\)logae
- \(\frac{d}{d x}\)(ex) = ex
- \(\frac{d}{d x}\)(eax) = aeax
Basic properties of differentiation:
i) Derivative of a constant: If f(x) = c (constant)
\(\frac{d}{d x}\){f(x)} = \(\frac{d c}{d x}\) = 0
ii) Derivative of the product of a constant and a function:
If f(x) = cg(x) then,
\(\frac{d}{d x}\){f(x)} = \(\frac{d}{d x}\){cg(x)} = c\(\frac{d g}{d x}\)
iii) Derivative of the sum or difference of two functions: If f(x) = g(x) ± h(x) then,
\(\frac{d}{d x}\){f(x)} = \(\frac{d}{d x}\){g(x) ± h(x)} = \(\frac{d g}{d x}\) ± \(\frac{d h}{d x}\)
Example:
y = \(\frac{6 x^6+8 x^2-2}{x^3}\) or, y = \(\frac{6 x^6}{x^3}\) + \(\frac{8 x^2}{x^3}\) – \(\frac{2}{x^3}\)
or, y = 6x3 + 8x-1 – 2x-3
∴ \(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(6x3) + \(\frac{d}{d x}\)(8x-1) – \(\frac{d}{d x}\)(2x-3)
= 6 × 3x3-1 + 8(-1)x-1-1 – 2(-3)x-3-1
= 18x2 – 8x-2 + 6x-4
iv) Derivative of the product of two functions: If f(x) = g(x)h(x) then,
\(\frac{d}{d x}\){f(x)} = \(\frac{d}{d x}\){g(x)h(x)} = g\(\frac{d h}{d x}\) + h\(\frac{d g}{d x}\)
Example:
y = (3x – 7)(5 – 6x)
∴ \(\frac{d y}{d x}\) = (3x – 7)\(\frac{d}{d x}\)(5 – 6x) + (5 – 6x)\(\frac{d}{d x}\)(3x – 7)
= (3x – 7)(-6) + (5 – 6x)(3)
= -18x + 42 + 15 – 18x = -36x + 57
v) Derivative of the ratio of two functions:
If f(x) = \(\frac{g(x)}{h(x)}\) then,
\(\frac{d}{d x}\){f(x)} = \(\frac{d}{d x}\){\(\frac{g(x)}{h(x)}\)} = \(\frac{h \frac{d g}{d x}-g \frac{d h}{d x}}{h^2}\)
Example:
Chain rule of differentiation : If y = f(x) and x = g(z) then \(\frac{d y}{d z}\) = \(\frac{d y}{d x} \cdot \frac{d x}{d z}\) ….. (3)
Example:
y = u5 and u = x2 + 3
∴ \(\frac{d y}{d u}\) = 5u4 = 5(x2 + 3)4 and \(\frac{d u}{d x}\) = 2x
∴ \(\frac{d y}{d x}\) = \(\frac{d y}{d u} \cdot \frac{d u}{d x}\) = 5(x2 + 3)4 ᐧ 2x
= 10x(x2 + 3)4
We can also express y = f(x) as x = g(y). In that case, if the value of \(\frac{d y}{d x}\) is not 0 then from equation (3), we get
1 = \(\frac{d y}{d x} \cdot \frac{d x}{d y}\) or, \(\frac{d y}{d x}\) = \(\frac{\frac{1}{d x}}{d y}\)
Second order derivative: Second order derivative means, the derivative of the derivative of the function y = f(x) and it is written as
\(\frac{d}{d x}\)(\(\frac{d y}{d x}\)) or, \(\frac{d^2 y}{d x^2}\)
Example:
x = 3cosπt + 4sinπt
∴ \(\frac{d x}{d t}\) = 3(-sinπt)π + 4(cosπt)π
= -3πsinπt + 4πcosπt
∴ \(\frac{d^2 x}{d t^2}\) = -3π(cosπt) + 4π(-sinπt)π
= -π2(3cosπt + 4sinπt) = -π2x