Fundamental Rules for Differentiation Rule (I): Differentiation of a constant function is zero i.e., . Rule (II): Let f(x) be a differentiable function and let c be a constant. Then c.f(x) is also differentiable such that . This is the derivative of a constant times a function is the constant times the derivative of the Read more about Fundamental Rules for Differentiation[…]
Bayes Theorem In Probability, Bayes theorem describes the probability of an event, based on the prior knowledge of conditions that might be related to the event. If A₁, A₂, A₃, … An are n mutually exclusive events from the Sample Space S, B is any other event from S and if probability of occurrence of Read more about Bayes Theorem[…]
Division of Objects into Groups Division of items into groups of unequal size Number of ways in which (m + n) distinct items can be divided into two unequal groups containing m and n items is . The number of ways in which (m + n + p) items can be divided into unequal groups Read more about Division of Objects into Groups[…]
De-Arrangement Selection of objects from two or more groups: if there are n₁ objects of one kind, n₂ objects of second kind, … nk objects of kth kind, then the number of ways of selecting r objects out of these objects is equal to coefficient of x² in [(1 + x + x² + … Read more about De-Arrangement[…]
Derivative at a Point Let f(x) be a differentiable function on [a, b] then corresponding to each point c ϵ [a, b] we obtain a unique real number equal to the derivative f’(c ) of f(x) at x = c This correspondence between the points in and derivatives at their points defines a new real Read more about Derivative at a Point[…]
Standard Derivatives (i) Algebraic Functions: a) , . b) , . c) , . (ii) Exponential Functions: a) , . b) , . (iii) Logarithmic Functions: a) , . b) , . (iv) Trigonometric Functions: a) d/dx (sin x) = cosx b) d/dx (cos x) = – sin x c) d/dx (tan x) = sec²x Read more about Standard Derivatives[…]
Derivatives of Implicit functions If the variables x and y are connected by a relation of the form f(x, y) = 0 and it is not possible or convenient to express y as a function of x in the form y = ф(x), then y is said to be an implicit function of x. To Read more about Derivatives of Implicit Functions[…]
Differentiation by Substitution Methods Substitution is useful to reduce the function into simple form. For problems involving inverse trigonometric functions, first try for a suitable substitution to simplify it and then differentiate it. If no such substitution is found, then differentiate directly. Some standard substitutions are given below. S.No Expressions Substitutions 1 x = atanθ Read more about Differentiation by Substitution Methods[…]
Derivative of a Function (Chain Rule) and Composite Function Derivative of a Function (Chain Rule): If y is a differentiable function of t and t is a differentiable function of x i.e., y = f(t) and t = g(x) then . Similarly, if y = f(u ), where u = g(v) and v = h(x) then Read more about Derivative of a Function (Chain Rule) and Composite Function[…]
Conditional Probability The probability of B under the assumption that A takes place. It is denoted by P(B/A) and is called the conditional probability of B given that A takes place Sometimes the probability of a given event depends on the occurrence or non-occurrence of some other event. Suppose A and B are two events in a Read more about Conditional Probability[…]