Theorem 0.1. If a function f(x) is differentiable at a point x = c in its domain, then f(c) is continuous at x = c. 

Note that the converse is definitely not true, as, for example, f(x) = |x| is continuous at x = 0, but not differentiable there. Note also that, for a function f(x) to be continuous at x = c,we must have

 lim x→c f(x) = f(c)

But we can also write this as limx→c f(x) − f(c) = 0. 

This will be useful.

Proof. Assume we have a function f(x) that is differentiable at a point x = c in its domain. Then the limit

                                      

                                           

                                                                        f(x) − f(c)

                                         f' (c) = lim x→c     -------------

                                                                            x - c                          

exists. Knowing this, we calculate lim x→c f(x) − f(c) as follows (we would like it to be 0):

  

Let 

f$$�$$

 be the differentiable function at 

x=a$$�=�$$

.
Then according to the basic definition of the differentiation, Differentiation of a function is equals to

f′(c)=limx→af(x)−f(a)x−a$${ }^{}{\displaystyle \frac{\mathrm{<span\; style="font-family:\; "Open\; Sans",\; sans-serif;\; word-spacing:\; normal;"><br> </span>}}{}}$$

We know that the for a function to be continuous at a point it must satisfy the equation

 lim  x→a f(x)=f(a)$$\underset{ }{<br>}$$

We can write the above equation as

                       

⇒limx→a(f(x)−f(a))=0     ..........  (2)

     $$$$

Ours a perfect unique solar system ?

Why our solar system is so unique in universe, there are nine planets circling the sun in various orbits. Jupiter is the most important planet in our solar system , it protects the planets from asteroids particularly the inner most planets , Its very common to have binary star systems in many galaxies and we have seen many stars being orbited by a few exoplanets unlike our solar system ,  we have not seen any other solar system so unique as ours. Our solar system has 9 planets with proper distribution of heavy planets. Earth being at perfect  distance from sun to support life , The Jupiter being exactly at the half way. The rocky planets in the inner circle and big gas giants Neptune and Saturn at the outer. 

Sun is 100 times bigger than Earth in diameter and Jupiter is 11 times bigger than Earth, 

diameter for Earth of 12,756 km

diameter for Jupiter 139822 km

diameter for Sun is 1392000  km

dearth = 12756  

dJupiter =139822  

dSun = 1392000   

Volume 

Volume of a sphere is given by the formula 

V = 4/3 π r³

VEarth = (4/3) * 3.14 * (dearth/2) **3  = 1086230340743.0399

VJupiter = (4/3) * 3.14 * (dJupiter/2) **3 = 1430556211858396.5

VSun  =  (4/3) * 3.14 * (dSun/2) **3  =   1.41154947072e+18

Ration of volumes Jupiter and Sun with respect to Earth

1316.99 ,  1299493.68

More than 1300 Earths would fit inside Jupiter where as it would take approximately 1.3 million Earths to fill the Sun's volume.

Derivative Rules

Mathematically, the derivatives are used to find the slope of a line or slope of a curve. Derivative rules are used to find the derivatives of different operations and different types of functions such as Trigonometric functions, power functions, logarithmic functions, exponential functions, etc. The following are some of the important derivative rules   

Sum Rule 

ddx(f(x)+g(x))     =   f'(x)+g'(x)

Difference Rule

ddx(f(x)−g(x))   =  f'(x)−g'(x).

Power Rule

Let n$\mathrm{ n}$ be a positive integer. If f(x)=xn$$,then

f'(x)=nxn−1.

Chain Rule 

If y is function u and u is a function x , then

dydx=dydu⋅dudx.

Product Rule

ddx(f(x)g(x))   =  f'(x)g(x)+g'(x)f(x).

Quotinet Rule

 ddx(f(x)g(x))=ddx(f(x))⋅g(x)−ddx(g(x))⋅f(x)(g(x))2.                                 

                        =   

f'(x)g(x)−g'(x)f(x)(g(x))2.(Ai

Partial Derivatives 

Suppose, we have a function f(x, y), which depends on two variables x and y, where x and y are independent of each other. Then we say that the function f partially depends on x and y.

So, the partial derivative of f with respect to x will be ∂f/∂x keeping y as constant. Similarly the partial derivative of f with respect to y is ∂f/∂y keeping x constant.

fx = ∂f / ∂xfy = ∂f / ∂y second order derivative of x is given by fxx = ∂²f / ∂x² = ∂ / ∂x (∂f / ∂x) = ∂ / ∂x (fx)Second order derivative of  y is given by    fyy = ∂²f / ∂y² = ∂ / ∂y (∂f / ∂y) = ∂ / ∂x (fy)Similary the second order derivative of xy or yx is given byfxy = ∂²f / ∂y ∂x = ∂ / ∂y (∂f / ∂x) = ∂ / ∂y (fx) fyx = ∂²f / ∂x ∂y = ∂ / ∂x (∂f / ∂y) = ∂ / ∂x (fy)If y = f(x) is a function where x is again a function of two variables u and v (i.e., x = x (u, v)) then∂f/∂u = ∂f/∂x · ∂x/∂u;∂f/∂v = ∂f/∂x · ∂x/∂vThe chain rule ,quotient rule and partial derivatives  are extensively used in AI (Artificial Intelligence), particularly the chain rule for computing the weights using back propagation.