8.Chemistry & Mathematics IV-Calculus(1)-Differentiation(1).

 

Calculus.

After discussing about graphs now we shall discuss a somewhat esoteric mathematical topic namely Calculus which includes the following two types –

i)Differential Calculus – Differentiation , finding Derivatives.

ii)Integral Calculus – Integration.

In Chemistry, we come across derivatives and their integrals several times (eg. while studying thermodynamics,chemical kinetics), but most of the times we fail to understand the purpose of using these mathematical methods in our study.In this post , I intend to discuss the concept of derivatives and how they help us in our study of Chemistry.

So what is Calculus?

Calculus is the study of how quantities change. Here we study the ‘instantaneous’ changes i.e changes over small intervals of time. If our experimental data values do not show regular change i.e they do not show regular increase or decrease , then studying change over small intervals of time gives us more accurate understanding of the variation of our data.

For example, suppose we are studying the trends in the stock/equity market.In stock market the prices of shares go up and down so often, that there is no regular change in the prices.The concept of calculus is helpful for studying such fluctuating data.

Differentiation

The process of finding the derivative or rate of change of a function is called differentiation. Here, we basically break down the measurements in small time intervals.A differential equation represents the relationship between a continuously varying quantity and its rate of change.

Point to remember – 

Derivative = instantaneous rate of change of the quantity ⇒ breaking down measurements in small time intervals.

A differential equation contains one or more terms involving derivatives of one variable(dependent variable,y) with respect to another variable (independent variable , x).

e.g. dy/dx = 2x is a differential equation.
For an algebraic equation the solution/s are numbers or values.The solution for differential equations are functions or set of functions.

Functions are expressions defining the relationship between two different quantities.

e.g. y= x2 is a function relating x&y.The differential equation for this function is

y’ = f’(y)=  dy/dx = 2x

Notation for differentiation

First Order Derivative    ⇒ y’ , f’(y) or  dy/dx       mean the same thing

Second order Derivative ⇒y”, f”(y) or  d2y/dx2  mean the same thing 

The first derivative of a function denotes the change of a function. The second derivative denotes the change of the first derivative.
e.g. dx/dt     = velocity ( First derivative)⇒ rate of change of displacement ‘x’ w.r.t time.
d2x/dt2 = acceleration (second derivative)⇒rate of change of velocity’dx/dt’ w.r.t time.


                         
We can calculate derivative by a derivative calculator available online. The link is –http://www.derivative-calculator.net.  One can just plug in the value to find the first,second derivative and see the corresponding graph .We, as students of Chemistry, can very well use this for our convenience as we are not expected to know the mathematical details of this concept.

We need to know some basic derivative rules though and they are as follows –

1)Derivative of a constant is zero.

   e.g. If c is a constant then dc/dt  = 0 .
                                                           

2) If y is a function of the type y = xn then,

     y’ = f’(y) = dy/dx = n x (n-1).

   e.g.  If y = x3 then y’ = 3x2 .
If y= x -3 then y’ = -3x-4.

3) Addition rule

If y=f(x) + g(x) then y’=f’(x) + g’(x) i.e  d/dx  [f(x) +g(x)] = df/dx + dg/dx .
                                                                           

4)Multiplication rule

If y=f(x).g(x) then y’=f’(x).g(x) + f(x).g’(x) and
If y= c. f(x) where c= constant then,
    y’= c. f’(x)  i.e    y’=dy/dx = c .df/dx.

5) If y = ln x then dy/dx = 1/x

6) If y = ex then dy/dx =  ex .

 7)If y = sin x then y’ = cos x
     If y = cosx then y’ = – sinx
If y= tan x then y’ = 1/cos2x .

In my next post we shall try to understand in greater depth the concept of derivatives. Till then ,

Be a perpetual student of life and keep learning …

Good day!

References and Further Reading –

1)http://www.maths.surrey.ac.uk/explore/vithyaspages/differential.html

2)http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/slope.html

3)http://www-math.mit.edu/~djk/calculus_beginners/chapter01/section02.html

4)https://www.math10.com/en/algebra/derivative-function.html

5)https://math.dartmouth.edu/opencalc2/cole/lecture8.pdf

6)https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/differential-equations-intro/v/differential-equation-introduction

 

 

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