1. Differentiation of Algebraic Functions :

(i) `d/(dx) x^n = nx^(n-1)`

(ii) `d/(dx) (sqrt x) = 1/( 2 sqrt x)`

(iii) `d/(dx) (1/x^n) =- n/(x^(n+1))`


2. Differentiation of Trigonometric Functions :

(i) `d/(dx) (sin x)= cos x`

(ii) `d/(dx) (cos x) =-sin x`

(iii) `d/(dx) (tan x) = sec^2 x`

(iv) `d/(dx) (sec x)= sec x tan x`

(v) `d/(dx) (cot x) =-cosec^2 x`

(vi) `d/(dx) (cosec x) =-cosec x cot x`


3. Differentiation of Logarithmic and Exponential Functions :

(i) `d/(dx) (log x) =1/x` , for `x > 0`

(ii) `d/(dx) (e^x) =e^x`

(iii) `d/(dx) (a^x) = a^x log a `, for `a > 0`

(iv) `d/(dx) (log_a x)= 1/(x log a) ` for ` x > 0, a > 0 , a ne 0`


4. Differentiation of Inverse Trigonometric Functions :

Sometimes the given function can be deducted with the help of inverse trigonometrical substitution and then to
find the differential coefficient is very easy.

(i) `d/(dx) (sin^(-1) x) = 1/(sqrt (1-x^2))` for `-1 < x < 1`

(ii) `d/(dx) (cos^(-1) x)= -1/(sqrt (1-x^2)) ` for `-1 < x < 1`

(iii) `d/(dx) (sec^(-1) x) =1/(|x| sqrt(x^2 -1))` for `|x| > 1`

(iv) `d/(dx) (cosec^(-1) x) =- 1/(|x| sqrt(x^2 -1))` for `|x| > 1`

(v) `d/(dx) (tan^(-1) x) =1/(1+x^2)` for `x in R`

(vi) `d/(dx) (cot^(-1) x)=-1/(1+x^2)` , for `x in R`

If the relation between the variables `x` and `y` are given by an equation containing both the variables and this equation is
not immediately solvable for `y`, then `y` is called an implicit function of `x` .lmplicit functions are given by `f(x, y)= 0`. To
find the differentiation of implicit function, we differentiate each term w.r.t. `x` considering `y` as a function of `x` and then
collect the terms of `dy//dx` together on left hand side and remaining terms on right hand side and find `dy// dx`.

Algebra of differentiation is defined in following ways:

1. Differentiation of the Sum of Two Functions :

Let `f(x)` and `g(x)` be two real valued functions. Then, `[f(x) + g(x)]' = f'(x) + g'(x)`

2. Differentiation of the Difference of Two Functions :

Let `f(x)` and `g(x)` be two real valued functions. Then, `[f(x) - g(x)]' = f'(x)- g'(x)`

3. Differentiation of the Product of Two Functions :

Let `f(x)` and `g(x)` be two real valued functions. Then, `[f(x) * g(x)]' = f'(x) g(x) +f (x)* g'(x)`

Note : If three functions are given, then `d/(dx) [f(x) * g(x) * h(x) ]= f(x) * g(x) * h'(x) +f(x) * g'(x) * h(x)+ f'(x) * g(x) * h(x) =((fg)'h+(gh)'f+ (hf)' g)/2`

4. Differentiation of the Quotient of Two Functions

Let `f(x)` and `g(x)` be two real valued functions. Then `[(f(x))/(g(x))]' =(f'(x) * g(x) -g'(x) * f(x))/(|g(x)|^2)` provided `g(x) ne 0`

while applying the quotient rule, think twice and check whether the given function could be simplified prior to
differentiation

5. Chain/Composite Rule :

lf `y=f{g(x)},` then `(dy)/(dx)=f'{g(x)}g'(x)`

A relation expressed between two variables `x` and `y` in the form `x = f(t ), y = g(t)` is said to be parametric form, where t
is a parameter. The derivative `dy// dx` of such function is given by

`(dy)/(dx) =(dy//dt)/(dx//dt)`

` (dy)/(dx) =(g'(t))/(f'(t))` provided `f'(t) ne 0`.

When a function consists of product or quotient of number of functions, then we take the logarithm and then
differentiate.

The functions which can be evaluated by using this method are of following types

(i) `y= {f_1 (x)} ^({f_2 (x)})`

(ii) `y= f_1(x) , f_2(x) , f_3 (x)........`

(iii) `y= (f_1(x) * f_2(x) * f_3(x).........)/(phi_1(x) * phi_2(x) * phi_2(x).........)`

Note : If `y = {f (x)} ^(g(x))`, then `(dy)/(dx) =` Differentiation of `{f(x)}^(g( x))` w.r.t. `x`

[taking `g(x)` as a constant] + Differentiation of `{f(x)* g(x)` w.r.t. ` x` [taking `f(x)` as a constant]

`(dy)/(dx) =g(x) * {f(x)}^(g(x)-1) * d/(dx) f(x) +{ f(x) }^(g(x)) * log f(x) * d/(dx) g(x)`

lf `y=f(x)` and `z=g(x)`

`=> (dy)/(dz)=(dy//dx)/(dz//dz) =(f'(x))/(g'(x))`

If `y= |(p,q,r),(u,v,w),(l,m,n)|` then

`(dy)/(dx) = |((dp)/(dx) , (dq)/(dx) , (dr)/(dx)),(u,v,w), (l,m,n)| + | (p, q, r), ((du)/(dx) , (dv)/(dx) , (dw)/(dx)), (l,m,n)| +| (p,q,r), (u,v ,w), ((dl)/(dx), (dm)/(dx) , (dn)/(dx))|`

For given `y = f(x )`, the process of finding its higher derivatives is called successive differentiation. The derivative of order two is denoted by `(d^2y)/(dx^2)` or `y_2 ` or `y''` and is obtained by differentiating `dy // dx` again w.r.t. `x` or differentiating `y = f(x)` twice w.r.t. `x`.

Note : The derivative of order three is obtained by differentiating `(d^2 y)/(dx^2)` again w.r.t. `x` and it is denoted by `(d^3y)/(dx^3)` or `y_3` or `y'''`

In order to find second order derivative of parametric function, we can also use the following formulae

Let `x= phi (t), y= psi (t)`

Then , `(d^2y)/(dx^2) = d/(dx) ((dy)/(dx))`

`= d/(dx) ((phi'(t))/(phi'(t)))`

`=(d/(dt) ((psi '(t))/(phi '(t))))/((dx)/(dt))`