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# Differentiation Formulas

Top
 Sub Topics Differentiation Formulas are useful in finding a derivative of Functions and we use them in solving differentiation problems. Here, we will discuss useful differentiation formulas and other properties:- Derivative of constant is 0:  d (c) = 0              dx Derivative of function with a constant is: d (a.f(x)) = a.f1(x)             dx Derivative of addition/subtraction between two Functions: d (f(x) + g(x)) = f1(x) + g1(x)  dx Derivative of xn is : d (xn) = n.xn-1 dx Now we will see the different types of Differential Equations. Some of the differential equations are shown below: First order differential equation: Different equations are combined in the first order differential equation.  The different types are mention here: Linear equation: An equation which is used to identify and explain the linear first order differential equations is known as linear equation. Separable equation: A separable equation is used to classify and assess separable first order differential equations. Exact equation: It is used to isolate and find the exact differential equation. Bernoulli differential equation: Bernoulli differential equation is one of important equations in the differential. In the Bernoulli differential equation substitution method is used to solve the differential equations. Methods to solve first order differential equations: Substitution: This method is also used to solve the differential equation. Euler’s method: It is also one of the important methods in the differentials; it is used to solve the approximation of solutions of differential equations.   These all are the first order differential equation. Some rules for the differentiation are also defined for the differentiation equations which are shown below: Rule 1: As we know that the differentiation of a constant function is always zero. Rule 2: In differentiation, the derivative of constant times a function. Rule 3: Product rule: Let f (p) and g (p) are the differentiable function, then the function f (p) and g (p) is also differentiable function.

## Basic Formulas

Suppose we have two function of 'x' that is 'u' and 'v', where 'a' and 'n' are constants, and 'd' is the differential operator. Let’s see basic general rule:

Linearity   rule: (d/dx) (a u) = a (du/dx)

Addition   rule: (d/dx) (u+v) = du/dx   + dv/dx

Subtraction rule: (d/dx) (u -v) = du/dx – dv/dx

Product rule: (d/dx) (u *v)=  u dv/dx + v du/dx

Quotient rule :( d/dx) (u/v) = (v du/dx – u dv/dx)/v2

Let’s see the basic function based formula:

Basic Functions :( d/dx) a =0

(d/dx) x=1

(d/dx) x= n x n-1

(d/dx) |x| = x/|x|, x! =0

(d/dx) e =e x

(d/dx) a= (ln a) ax   (a>0)

(d/dx) ln x = 1/x

Trigonometry function:

(d/dx) sin x =cos x

(d/dx) cos x   = -sin x

(d/dx) tan x = sec2 x

(d/dx) cot x = -cosec x

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

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

(d/dx) arcsin x = sin-1 x =1/√ (1-x2)

(d/dx) arccos x = cos-1 x =-1/√ (1-x2)

(d/dx) arctan x = tan-1 x =1/ (1 +x2)

(d/dx) arccot x = cot-1 x =-1/ (1+x2)

(d/dx) arcsec x = sec-1 x =1/ [|x|√ (x-1)]

(d/dx) arccosec x = cosec-1 x =-1/ [|x|√ (x-1)]

## Trigonometry formulas

Trigonometry is a mathematical branch in which we study about triangle and their relationships with sides and  angle. In Trigonometry, some basic Functions exist with respect to triangle  -

• sin x   =    Opposite
hypotenuse
hypotenuse
• tan x  =     Opposite
• cosec x   =    Hypotenuse
Opposite
• sec x   =    Hypotenuse
Opposite

here x represents angle between hypotenuse and adjacent of triangle .

There are some basic trigonometric formulas, which shows relationship between these trigonometric Functions :

sin2x + cos2x  = 1

sec2x - tan2x  = 1

cosec2x - cot2x  = 1

For differentiation we use  some  basic  trigonometry Differentiation Formulas -

1.  d   (sin  x)          =  cos x
dx
2. d   (cos  x)          =  -sin x
dx
3. d   (tan  x)          =  sec2 x
dx
4. d   (cot  x)          = - cosec2 x
dx
5. d   (sec  x)          =  sec x . tan x
dx
6. d   (cosec  x)          =  -cosec x .  cot  x
dx
7. d   (sin-1  x)          =    1
dx                               √1-x2
8. d   (cos-1  x)          =    -1
dx                               √1-x2
9. d   (tan-1  x)          =    1
dx                               1+x2
10. d   (ex)          =   ex
dx
11. d   (ax)          =   ax . ln a
dx

## Hyperbolic trigonometry formulas

Hyperbolic Functions are analog of the trigonometric functions. The basic Hyperbolic Functions are the hyperbolic "sinh" and the hyperbolic cosine "cosh" from which, several other functions are derived like hyperbolic tangent "tanh", and many more derived trigonometric functions. Hyperbolic trig functions, are defined using ex and e–x.  Hyperbolic Trigonometry plays an important role when trigonometry functions have imaginary or complex problems. Hyperbolic functions were introduced in the 1760s by Byich Lambert Vincenzo Riccati and Johann Heinr. The formulas for Hyperbolic Trigonometry functions are as follows:

Sinh(x)= (ex-e-x)/2.

Csch(x)=1/sinh(x)=2/(ex-e-x).

Cosh(x)=(ex+ e-x)/2.

Sech(x)=1/cosh(x)=2/(ex+ e-x).

Tanh(x)=sinh(x)/cosh(x)=(ex- e-x)/(ex + e-x).

Coth(x)= 1/tanh(x)=(ex+ e-x)/(ex-e-x).

Hyperbolic Trigonometric Functions are given as:

Cosh2(x)- sinh2(x)=1.

Tanh2(x) + sech2(x)=1.

Coth2(x) – csch2(x)=1.

Relations to trigonometric functions of hyperbolic trigonometry are given with help of following formulas:

Sinh(z) = - isin (iz).

Csch(z)=  i csc(iz).

cosh(z)= cos(iz).

sech(z) = sec(iz).

tanh(Z) = - I tan (iz).

coth(z) = I cot(iz).

For the hyperbolic trigonometry, we can say that the cosh function is even, while the sinh function is odd i.e for the pythagoras theorem.

(coshx)2 – (sinh x)2=1

These are the formula’s using which we can easily solve problems related to hyperbolic trigonometry. Hyperbolic functions occur in the solutions of some important linear Differential Equations. It is also used in many areas of physics including electromagnetic theory, heat transfer, etc.