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
Derivative of function with a constant is:
d (a.f(x)) = a.f1(x)
Derivative of addition/subtraction between two Functions:
d (f(x) + g(x)) = f1(x) + g1(x)
Derivative of xn is :
d (xn) = n.xn-1
Basic FormulasBack to Top
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) xn = n x n-1
(d/dx) |x| = x/|x|, x! =0
(d/dx) e x =e x
(d/dx) ax = (ln a) ax (a>0)
(d/dx) ln x = 1/x
(d/dx) sin x =cos x
(d/dx) cos x = -sin x
(d/dx) tan x = sec2 x
(d/dx) cot x = -cosec 2 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|√ (x2 -1)]
(d/dx) arccosec x = cosec-1 x =-1/ [|x|√ (x2 -1)]
Trigonometry formulasBack to Top
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
cos x = Adjacent
tan x = Opposite
cosec x = Hypotenuse
sec x = Hypotenuse
cot x = Adjacent
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 -
d (sin x) = cos x
d (cos x) = -sin x
d (tan x) = sec2 x
d (cot x) = - cosec2 x
d (sec x) = sec x . tan x
d (cosec x) = -cosec x . cot x
d (sin-1 x) = 1
d (cos-1 x) = -1
d (tan-1 x) = 1
d (ex) = ex
d (ax) = ax . ln a
Hyperbolic trigonometry formulasBack to Top
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:
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:
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).
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.