*Here's a very quick outline or summary of the concepts introduced in this tutorial. Future tutorials, which appear further in this series of calculus tutorials, will focus on how these ideas are applied and we will solve interesting examples and problems where these concepts will be applied.*

**• Diﬀerentiability Of Functions & Basic Diﬀerentiation Formulas: **

The condition for differentiability of functions will be introduced in the tutorial. Addition, Subtraction, Linearity, Product, Quotient, Chain and Power Rules, Exponential and Logarithmic Rules - these will be introduced in the tutorial document.

Formulas for derivatives of* sin x, cos x, tan x, sec x, cot x, cosec x and their inverse values: sin*^{-1} x, cos^{-1}^{ }x, tan^{-1}^{ } x, sec^{-1}^{ } x, cot^{-1}^{ } x, cosec^{-1}^{ } x

**• Successive Diﬀerentiation: **

The derivative f' (x) of a derivable function f (x) is itself a function of x. We suppose that it also possesses a derivative, which is denoted by f'' (x) and called the second derivative of f (x). The third derivative of f (x) which is the derivative of f'' (x) is denoted by f '''(x) and so on. Thus the successive derivatives of f (x) are represented by the symbols, f (x), f; (x), . . . , f ^{n} (x), . . .

where each term is the derivative of the previous one. Sometimes y_{1} , y_{2} , y_{3} , . . . , y_{n} , . . . are used to denote the successive derivatives of y.

**• Leibnitz’s Theorem**

The nth derivative of the product of two functions: If u, v be the two functions possessing derivatives of the nth order, then (uv)_{n} = u_{n} v +^{n }C_{1 }u_{n−1 }v1 +^{n} C_{2} u_{n−2 }v2 + . . . +^{n} C_{r} u_{n−r }v_{r }+ . . . + uv_{n} .

**• Rolle’s Theorem: **

If a function f (x) is derivable in an interval [a, b], and also f (a) = f (b), then there exists atleast one value c of x lying within [a, b] such that f (c) = 0.

**• Lagrange’s Mean Value Theorem: **

If a function f (x) is derivable in an interval [a, b], then there exists atleast one value c of x lying within [a, b] such that

(f (b) − f (a)) / (b-a) = f(c)

**Increasing and Decreasing Functions:**

A function whose derivative is positive for every value of x in an interval **is a monotonically increasing function **of x in that interval, i.e,

If f (x) > 0 for every value of x in [a, b], then f (x) is an increasing function of x in that interval.

A function whose derivative is negative for every value of x in an interval **is a monotonically decreasing function** of x in that interval, i.e,

If f (x) < 0 for every value of x in [a, b], then f (x) is an decreasing function of x in that interval.

**What Maxima and Minima mean :**

Maximum Value of a function: f (c) is said to be a maximum value of f (x), if it is the greatest of all its values of x lying in some neighbourhood of c, i.e, f (c) is a maximum value of x if there exists a positive δ such that f (c) > f (c + h) or f (c) − f (c + h) > 0 for values of h lying between −δ and δ.

Minimum Value of a function: f (c) is said to be a minimum value of f (x), if it is the least of all its values of x lying in some neighbourhood of c, i.e, f (c) is a minimum value of x if there exists a positive δ such that f (c) < f (c + h) or f (c + h) − f (c) > 0 for values of h lying between −δ and δ.

**Greatest and least values of a function in any interval**

The greatest and least values of f (x) in any interval [a, b] are either f (a) and f (b), or are given by the values of x for which f (x) = 0.

Change of Sign

A function is said to change sign from positive to negative as x passes through a number c, if there exists some left-handed neighbourhood (c − h, c) of c for every point of which the function is positive, and also there exists some right-handed neighbourhood (c, c + h) of c for every point of which the function is negative.

Suﬃcient Criteria for extreme values

Prove that f (c) is an extreme value of f (x) if and only if f (x) changes sign as x passes through c, and to show that f (c) is a maximum value if the sign changes from positive to negative and a minimum value if the sign changes from negative to positive.

**Minimum and Maximum Values :**

**Minimum Value:** f (c) is a minimum value of f (x), if f '(c) = 0 and f'' (c) > 0.

**Maximum Value:** f (c) is a maximum value of f (x), if f '(c) = 0 and f ''(c) < 0.

**Implicit Diﬀerentiation: **

A relation F (x, y) = 0 is said to deﬁne the function y = f (x) implicitly if, for x in the domain of f , F (x, f (x)) = 0. Given a diﬀerentiable relation F (x, y) = 0 which deﬁnes the diﬀerential function y = f (x), it is usually possible to ﬁnd the derivative f even in the case when you cannot symbolically ﬁnd f . The method of ﬁnding the derivative is called implicit diﬀerentiation.

Criteria for concavity, convexity and inﬂexion:

Criteria to determine whether a curve y = f (x) is concave upwards, concave downwards, or has a point of inﬂexion at P [c, f (c)] are:

(i) the curve is concave upwards at P if f'' (c) > 0.

(ii) the curve is concave downwards at P if f'' (c) < 0.

(iii) the curve has inﬂexion at P if f'' (c) = 0 and f''' (c) = 0.

**Complete Tutorial with Definitions and Formulas :**