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MATHEMATICS FOR PHYSICS https://byjus.com/maths/differentiation-formulas/
Differentiation:
Constant quantity:
If the value of a quantity remains the same in a mathematical operation, it is called a constant quantity.
Examples: Integers (4, 7, 13, ...), the fraction (1/2, 4/5), π, e, etc.,
Variable quantity:
If the quantity takes different values in a mathematical operation, it is a variable quantity. Examples:
(i) In the equation y = 2x + 3, x and y are variables. (ii) In the equation F = ma where F, m and a are variables.
Variable quantities are divided into two types
(a) Independent quantity. (b) Dependent quantity.
In the equation y = 2x + 3, x is the independent quantity, and y is the dependent quantity.
Function:
If corresponding to any given value of x, there exists a single definite value then y is called a function of x. This is represented as y = f(x). This equation means that corresponding to one value of x, there is a single definite value of the variable y. Example: (i) Let y = 2x, when x = 1 then y = 2; when x = 2 then y = 4... Thus corresponding to each value of x, there is a definite value of y.
Difference and Differential coefficient:
Let y be a function of x. This is denoted as y = f(x). Here x and y are the variables. Let the value of x change to x+△x and correspondingly the value of y changes to y+△y.
The change between the initial and final values of a variable quantity is called Difference.
Difference in x = (x+ △x) - x = △x; Difference in y =(y+△y)-y=△y
The ratio of △y/△x is called the quotient of the two increments. When Difference in x (i.e., △x) is very small i.e., almost approaching to zero, then
we write △y/△x is equal to dy/dx.
In mathematical language, we represent the above statement as:
lim(∆x→0)█(∆y/∆x) =dy/dx or
Lt(∆x→0)█(∆y/∆x) =dy/dx
In this equation, ∆y/∆x is the ratio of a small quantity ∆y to another small quantity ∆x. But dy/dx is a single quantity and is called the differential coefficient of y with respect to x.
dy/ dx = Lt(∆x→0)█(∆y/∆x) = Lt(∆x→0)█[f(x+∆x)-f(x)]/∆x
The rate of change of a dependent variable with respect to the independent variable is called the differential coefficient or derivative.
d/dx does not mean that d is divided by dx. It is a single operator called the differential operator.
Differentiation:
The process of finding the differential coefficient of a function is called differentiation.
Basic Theorems on Differentiation: i) The derivative of a constant is zero. Let y = f(x) = c. Where c is constant.
Then dy/dx =d/dx(c) =0
Example: Differentiate y = 2a where a is a constant dy/dx= d/dx(2a) =0
ii) The differential coefficient of xn is obtained by decreasing the power of x by unity and multiplying by n.
If y = xn then dy/dx = nxn-1, where n may be positive or negative.
Example: Differentiate y = 8x8 w.r.t. x.
Solution: dy/dx = d/dx(8x8) = 8d/dx(x8) = 8*8*x8-1 = 64x7
iii) The derivative of the algebraic sum of two functions is equal to the algebraic sum of the derivatives of the two functions.
Let y = u±v±w±..... where u, v, w..... are all functions of x.
Then dy/dx = d/dx(u±v±w±.....) = d/dx(u)±d/dx(v)±d/dx(w)± ...
Example: Differentiate y = 3x4 + 2x2 - 10x w.r.t. x.
Solution:
dy/dx = d/dx(3x4 + 2x2 -10x) = d/dx(3x4) + d/dx(2x2)- d/dx (10x)
= 4.3.x4-1 + 2.2.x2-1 -10.x1-1 = 12x3 + 4x - 10
Rules of differentiation
Product Rule: The differential coefficient of the products of two functions = 1st function x differential coefficient of the 2nd function + 2nd function x differential coefficient 1st function.
Let y = uv where u and v are functions of x. Then dy/dx= d/dx(uv) = u(dv/dx) + v(du/dx)
Example: Differentiate y = x(x2 – 2x) w.r.t. x.
Solution: Here u = x, v = x2 - 2x
dy/dx = d/dx[x(x2-2x)]=x(d/dx)(x2 – 2x) + (x2–2x)d/dx(x). = x(2x-2)+(x2 - 2x)*1= 3x2 - 4x
Quotient Rule:
The differential coefficient of the quotient of two functions = [2nd function * derivati of the 1st function - 1st function * derivative of 2nd function] divided by the square of the second function. Let y = u/v where u and v are two functions of x. Then, dy/dx= d/dx[(x2 + 1)/(x-1)] w.r.t.x.
Example: Differentiate y = (x2 + 1)/(x - 1) w.r.t. x. Solution: Here u = x2 + 1, v = x - 1. range . [(x2 +1)/(x-1)7=(x-1) dx (x +1)-(x2 +1) & (x - 1) (x-1)2 (x-1)× 2x -(x +1)x1 x? - 2x - 1 (x-1) (x-1)
== MATHEMATICS FOR PHYSICS
‘Chanakya’ Today's Eenadu Article
Financial success is yours with these 5 Chanakya principles!
‘Chanakya’... Probably a name that does not want to be introduced in India. Governance, political maneuvers, management. With his help, Chandragupta Maurya wiped out the Nanda Empire. Later a special chapter in the history of India was written around the beginning of the Mauryan Empire.
Chanakya's principles for success in life are still taught as lessons in the name of ‘Chanakya ethics’. Books are being molded. He also taught many things to achieve financial success in personal life. Let's take a look at some of them.
Wealth in the right way
"Wealth must be kept in the right hands. Do not give to others under any circumstances. Sea water that turns into clouds always gives sweet fruit - Chanakya Its growth depends on how you manage the money. According to Chanakya, if the water turns into clouds, it will return to good form in the form of rain. Turns salty when mixed in the same sea.
So if you invest your money in the right way it will give good returns. That is why you should invest in safe stocks, bonds, mutual funds including gold, and reliable ones like NPS. The same thing applies with ezines and newsletters.
Success is the goal ‘‘ Ask yourself three questions before you start a task. Why are we doing that? What are the results? Will success pay off? Think deeply about these and move on if you find a satisfactory answer. Those who do not set a goal will not succeed - Chanakya
Do not be afraid of temporary losses
"When you start a business, do not worry about losing. Do not leave it in the middle under any circumstances. Those who work honestly are always happy - Chanakya
As with all things in life, there must be a definite goal in terms of investment. Or where to invest? How much to do? How Long Should I Invest? There is no clarity on the issues. The answer to the above questions can be found if you can set a goal for your investment. Knowing how much to earn at any given time will also make it easier to know where to invest. The stock markets have undergone many corrections over the last 30 years. Disasters, scandals, economic slowdown, etc. have contributed to this. However, no matter how much the indices fell .. they recovered and set new records.
Many investors panic during market corrections. They eventually withdraw their investments when they are at a loss. This converts the losses on paper into real losses. But, those who were stable at that time later got many times more revenue.
Extreme misery
"Extreme pride led to the death of Ravana. The emperor of Bali was in trouble due to his over-generosity. So anything insane. Stay away from it - Chanakya
Chanakya made it clear that he was not overly showy in any work. This also applies to investments. Investing too much in any single investment path can have negative consequences. That is why we need to diversify our investment. There are many benefits to this. Not all investment avenues offer the same returns in all cases. If it brings some losses .. some will bring huge gains. Doing the same diversification can reduce losses.
Learning from the mistakes of others
‘‘ Learn from the mistakes of others. One life is not enough to learn all the mistakes yourself - Chanakya
We learn a lot from experiences, especially from mistakes. But, the time was worth it. So in order to learn quickly and quickly one must also learn lessons from the mistakes of others. Many successful investors and financial experts bring their mistakes and experiences in book form. You can learn many things by reading them. You can also access information on authentic websites. In particular, the steps taken by our relatives and friends should be closely monitored.