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### Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Calculus AB, AP Calculus BC, A Level, Singapore/GCE A-Level; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!

Theory and definitions introducing differentiability, basic differentiation formulas of common algebraic and trigonometric functions , successive differentiation, Leibnitz Theorem, Rolle's Theorem,  Lagrange's Mean Value Theorem, Increasing and decreasing functions, Maxima and Minima; Concavity, convexity and inflexion, implicit differentiation.

### • 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.

### • 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 y1 , y2 , y3 , . . . , yn , . . . 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 = un v +n C1 un−1 v1 +n C2 un−2 v2 + . . . +n Cr un−r vr + . . . + uvn .

### • 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.

### In Case you'd like to take a look at some of our other tutorials related to Single Variable Calculus :

 Quick and introductory definitions related to Funtions, Limits and Continuity - Defining the domain and range of a function, the meaning of continuity, limits, left and right hand limits, properties of limits and the "lim" operator; some common limits;  defining the L'Hospital rule, intermediate and extreme value theorems. Functions, Limits and Continuity - Solved Problem Set I - The Domain, Range, Plots and Graphs of Functions;  L'Hospital's Rule- -  Solved problems demonstrating how to compute the domain and range of functions, drawing the graphs of functions, the mod function, deciding if a function is invertible or not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital rule. Functions, Limits and Continuity - Solved Problem Set II - Conditions for Continuity, More Limits, Approximations for ln (1+x) and sin x for infinitesimal values of x   - More advanced cases of evaluating limits, conditions for continuity of functions, common approximations used while evaluating limits for ln ( 1 + x ), sin (x); continuity related problems for more advanced functions than the ones in the first group of problems (in the last tutorial). Functions, Limits and Continuity - Solved Problem Set III - Continuity and Intermediate Value Theorems - Problems related to Continuity, intermediate value theorem. Introductory concepts and definitions related to Differentiation - Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc - Theory and definitions introducing differentiability, basic differentiation formulas of common algebraic and trigonometric functions , successive differentiation, Leibnitz Theorem, Rolle's Theorem,  Lagrange's Mean Value Theorem, Increasing and decreasing functions, Maxima and Minima; Concavity, convexity and inflexion, implicit differentiation. Differential Calculus - Solved Problem Set I - Common Exponential, Log , trigonometric and polynomial functions - Examples and solved problems - differentiation of common algebraic, exponential, logarithmic, trigonometric and polynomial functions and terms; problems related to differentiability . Differential Calculus - Solved Problem Set II - Derivability and continuity of functins - Change of Indepndent Variables - Finding N-th Derivatives - Examples and solved problems - related to derivability and continuity of functions; changing the independent variable in a differential equation; finding the N-th derivative of functions Differential Calculus - Solved Problems Set III- Maximia, Minima, Extreme Values,  Rolle's Theorem - Examples and solved problems - related to increasing and decreasing functions; maxima, minima and extreme values; Rolle's Theorem Differential Calculus - Solved Problems Set IV - Points of Inflexion, Radius of Curvature, Curve Sketching -  Examples and solved problems - Slope of tangents to a curve, points of inflexion, convexity and concavity of curves, radius of curvature and asymptotes of curves, sketching curves Differential Calculus - Solved Problems Set V - Curve Sketching, Parametric Curves - More examples of investigating and sketching curves, parametric representation of curves Introducing Integral Calculus - Definite and Indefinite Integrals - using Substitution , Integration By Parts, ILATE rule  - Theory and definitions. What integration means, the integral and the integrand. Indefinite integrals, integrals of common functions.  Definite integration and properties of definite integrals; Integration by  substitution, integration by parts, the LIATE rule, Integral as the limit of a sum. Important forms encountered in integration. Integral Calculus - Solved Problems Set I - Basic examples of polynomials and trigonometric functions, area under curves - Examples and solved problems - elementary examples of integration involving trigonometric functions, polynomials; integration by parts; area under curves. Integral Calculus - Solved Problems Set II - More integrals, functions involving trigonometric and inverse trigonometric ratios - Examples and solved problems - integration by substitution, definite integrals, integration involving trigonometric and inverse trigonometric ratios. Integral Calculus - Solved Problems Set III - Reduction Formulas, Using Partial FractionsI- Examples and solved problems - Reduction formulas, reducing the integrand to partial fractions, more of definite integrals Integral Calculus - Solved Problems Set IV - More of integration using partial fractions, more complex substitutions and transformations - Examples and solved problems - More of integrals involving partial fractions, more complex substitutions and transformations Integral Calculus - Solved Problems Set V- Integration as a summation of a series - Examples and solved problems - More complex examples of integration, examples of integration as the limit of a summation of a series Introduction to Differential Equations and Solved Problems - Set I - Order and Degree, Linear and Non-Linear Differential Equations, Homogeneous Equations, Integrating Factor -  Theory and definitions. What a differential equation is; ordinary and partial differential equations; order and degree of a differential equation; linear and non linear differential equations; General, particular and singular solutions; Initial and boundary value problems; Linear independence and dependence; Homogeneous equations; First order differential equations; Characteristic and auxiliary equations. Introductory problems demonstrating these concepts. Introducing the concept of Integrating Factor (IF). Differential Equations - Solved Problems - Set II - D operator, auxillary equation, General Solution - Examples and solved problems - Solving linear differential equations, the D operator, auxiliary equations. Finding the general solution ( CF + PI ) Differential Equations - Solved Problems - Set III - More Differential Equations - More complex cases of differential equations. Differential Equations - Solved Problems - Set IV - Still more differential equations.