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Edited By Vishal kumar | Updated on Sep 25, 2024 12:47 PM IST

The process of finding the derivative or the differential coefficient of a function w.r.t. the variable, on which it depends is called differentiation. Integration is the process of finding the function, whose derivative is given. for this reason, the process of integration is called the inverse process of differentiation.

In this article, we will cover the mathematical tools used in kinematics. These tools are crucial for studying any chapter, especially in the mechanics section, where a solid understanding of basic mathematics is essential for solving problems.

So let’s read the entire article to know in-depth about the differentiation and integration to use in kinematics to make the calculation easier.

Differentiation

Differentiation is very useful when we have to find rates of change of one quantity compared to another.

  • If y is one quantity and we have to find the rate of change of y with respect to x which is another quantity

Then the differentiation of y w.r.t x is given as dydx

  • For a y V/s x graph

We can find the slope of the graph using differentiation

I.e Slope of yV/s× graph =dydx
- Some important Formulas of differentiation
- ddx(xn)=nxn1

Example-

ddx(x5)=(n=5)dxn dx=nxn1dx5 dx=5x51dx5 dx=5x4

Similarly

ddxsinx=cosxddxcosx=sinxddxtanx=sec2xddxcotx=csc2xddxsecx=secxtanxddxcscx=cscxcotxddxex=exddxax=axlnaddxln|x|=1x

Integration

The opposite process of differentiation is known as integration.
Let x , and y be two quantities, using differentiation we can find the rate of change of y with respect to x , Which is given by dydx

But using integration we can get the direct relationship between quantities x and y

So let dydx=K where K is constant
Or we can write dy=Kdx

Now integrating on both sides we get the direct relationship between x and y

І.e dy=Kdxy=Kx+C

Where C is some constant

  • For a y V/s x graph

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We can find the area of the graph using the integration

Some Important Formulas of Integration

. xndx=xn+1n+1+C where (C = constant) E.g- xndx=,n=3xn+1n+1+Cx3+13+1+Cx44+Cdxx=ln|x|+Cexdx=ex+Caxdx=1lnaax+C. lnxdx=xlnxx+C

sinxdx=cosx+Ccosxdx=sinx+Ctanxdx=ln|cosx|+Ccotxdx=ln|sinx|+Csecxdx=ln|secx+tanx|+Ccscxdx=ln|cscx+cotx|+Csec2xdx=tanx+Ccsc2xdx=cotx+Csecxtanxdx=secx+Ccscxcotxdx=cscx+Cdxa2x2=sin1xa+Cdxa2+x2=1atan1xa+Cdxxx2a2=1asec1|x|a+C

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Solved Example Based on Differentiation and Integration

Example 1. Displacement of a particle as a function of time is given as s=at3+bt2+c. Then rate of Displacement as a function of time is
1) Independent of t
2) 3at
3) 2bt
4) 3at2+2bt

Solution:

The displacement of a particle as a function of time is given as
s=at3+bt2+c.

The rate of Displacement-
dsdt=ddt[at3+bt2+c]=3at2+2bt

Hence, the answer is option (1).

Example 2. If y=x3 then dydx is:
1) 3x2
2) 3x
3) 3
4) 3x3

Solution:

The formula of differentiation
ddx(xn)=nxn1

Given - y=x3
So using the formula we get
dydx=3x2

Hence, the answer is option (1).

Example 3. Then what is the relation between x and y then what is the relation between x and y
1) y=x33
2) y=x22
3) y=x44
4) y=x

Solution:

The formula of integration-

xndx=xn+1n+1+C where (C= constant )

Given-
dydx=x2dy=x2dxy=x2dxy=x33+c

As at x=0,y=0
so, y=x33
Hence, the answer is option (1).

Example 4. The distance travelled by an accelerated particle of mass M is given by the following relation s=6t+3t2. Then the rate of change of s after 2 seconds is
1) 18
2) 12
3) 6
4) 24

Solution:

As we learned
Some important Formulas of differentiation
ddx(xn)=nxn1

So s=6t+3t2
now using the formula
ds/dt=6+6t put t=(2sec)ds/dt=6+(6×2)=18

Hence, the answer is option (1).

Example 5. If sinθ=1/3, then find the value of cosθ
1) 89
2) 43
3) 223
4) 34

Solution:

cosθ=1sin2θ=1(1/3)2=8/9=223

Hence, the answer is the option (3).

Conclusion

The derivative or differential coefficient of a function is the limit to which the ratio of the small increment in the function to the corresponding small increment in the variable (on which it depends) tends to, when the small increment in the variable approaches zero. The concept of integration is used in physics to make measurements when a physical quantity varies in continuous manner.

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