Hello,

Application of derivatives are-

• To calculate the profit and loss in business using graphs.
• To check the temperature variation.
• To determine the speed or distance covered such as miles per hour, kilometre per hour etc.
• Derivatives are used to derive many equations in Physics.
• The rate of change of a quantity is the most important and general application of derivatives.

Thank you.

Hello student,

The gas law is the law that relates the pressure (P), volume (V), and temperature (T) of  a gas. There are three primary gas laws (Charles' Law , Boyle's Law and Avogadro's Law ) of gas that combines into a General Gas equation or Ideal Gas Law.

Formula : PV = nRT

• P is the pressure of the ideal gas.
• V is the volume of the ideal gas.
• n is the amount of ideal gas measured in terms of moles.
• R is the universal gas constant .
• T is the temperature.

Proof :

Let us consider the pressure exerted by the gas to be ‘ p, ’

The volume of the gas be – ‘ v ’

Temperature be – T

n – be the number of moles of gas

Universal gas constant – R

According to Boyle’s Law ,

At constant n & T, the volume bears an inverse relation with the pressure exerted by a gas.

i.e. v∝1p ………………………………(i)

According to Charles’ Law,

When p & n are constant, the volume of a gas bears a direct relation with the Temperature.

i.e. v∝T ………………………………(ii)

According to Avogadro’s Law,

When p & T are constant, then the volume of a gas bears a direct relation with the number of moles of gas.

i.e. v∝n ………………………………(iii)

Combining all the three equations, we have-

v∝nTp

or pv=nRT

where R is the Universal gas constant , which has a value of 8.314 J/mol-K

All the best and thank you.

hi,

you may use the differentiate formula,

dy/dx

suppose here y=e^3x,

formula for d[e^x]/dx is, e^x, now take the whole 3x as x , by using this formula you will get e^3x, here 3x is also a function of x, so we need to again differentiate it  with respect to x, the final result will be 3* e^3x.

Hello,

The given equation,

x^2+6xy+y^2

Differentiating the above equation

d/dx(x^2+6xy+y^2)=2x+6y

differentiating the above solution to get the second order derivative we get,

d/dx(2x+6y)=2

therefore, the second order derivative of the above equation that is, x^2+6xy+y^2 is 2.

Hope this helps.

All the best!