Mathematics
Continuity and Differentiability
सांतत्य तथा अवकलनीयता
This Class 12 Mathematics chapter develops the ideas of continuity and differentiability of functions. Students learn to test continuity at a point and on an interval, understand the relationship between continuity and differentiability, and apply standard rules for finding derivatives. The chapter covers derivatives of composite, implicit, inverse trigonometric, exponential, and logarithmic functions, along with logarithmic differentiation and derivatives of functions expressed in parametric form. It also introduces Rolle’s Theorem and the Mean Value Theorem.
1Continuity
सांतत्य
In this Class 12 Mathematics topic from Continuity and Differentiability, students learn how to determine whether a function is continuous at a point or over an interval. The topic connects limits with continuity through the conditions that the function value, limit, and one-sided limits must agree. Students examine graphs, identify removable and other discontinuities, test common algebraic and trigonometric functions, and solve problems involving continuity. These ideas provide the foundation for understanding differentiability and applying calculus confidently.
0 questionsDifferentiability
अवकलनीयता
In this Class 12 Mathematics topic from the chapter “Continuity and Differentiability,” students learn how differentiability describes the existence of a definite derivative at a point and what it reveals about a function’s local behaviour. The topic connects left- and right-hand derivatives, explains why differentiability implies continuity, and develops methods for testing differentiability of algebraic, trigonometric, exponential, logarithmic, and piecewise functions. Students also strengthen their use of standard derivative rules and apply these ideas to structured problems involving continuity and derivatives.
0 questionsDerivative of composite functions
संयुक्त फलनों का अवकलज
In Class 12 Mathematics, this topic from Continuity and Differentiability explains how to differentiate a composite function, written as f(g(x)), by applying the chain rule. Students learn to identify the inner and outer functions, differentiate each part in the correct order, and use the formula d/dx[f(g(x))] = f′(g(x))·g′(x). Worked examples with algebraic, trigonometric, exponential, and logarithmic expressions build accuracy in simplifying and checking derivatives.
0 questionsChain rule
संयुक्त फलनों के अवकलन का श्रृंखला नियम
In Class 12 Mathematics, the Chain Rule is studied in the chapter Continuity and Differentiability as a method for differentiating composite functions. Students learn to identify an inner function and an outer function, then find the derivative by multiplying the outer derivative by the derivative of the inner function: d/dx[f(g(x))] = f′(g(x))g′(x). The topic develops accuracy in differentiating algebraic, trigonometric, exponential, and logarithmic expressions and helps students solve step-by-step derivative problems.
0 questionsDerivatives of inverse trigonometric functions
प्रतिलोम त्रिकोणमितीय फलनों के अवकलज
In Class 12 Mathematics, this topic from the chapter Continuity and Differentiability explains how to differentiate inverse trigonometric functions such as sin⁻¹x, cos⁻¹x, and tan⁻¹x. Students learn the standard derivative formulas, understand the importance of domains and principal values, and apply the chain rule to composite expressions. Practice includes differentiating sums, products, and nested functions while maintaining correct notation, signs, and simplification.
0 questionsDerivatives of implicit functions
निहित फलनों के अवकलज
In this Class 12 Mathematics topic from Continuity and Differentiability, students learn how to differentiate equations in which the variables are not explicitly separated, such as x² + y² = a². They use implicit differentiation by treating y as a function of x, apply the chain rule, and find dy/dx and, where appropriate, higher-order derivatives. The topic also develops understanding of tangent slopes and differentiability for implicitly defined curves.
0 questionsExponential functions
घातांकीय फलन
In this Class 12 Mathematics topic from Continuity and Differentiability, students study functions in which the variable appears in the exponent, such as a^x for a positive base a. They learn their basic properties, domain, range, graphs, growth and decay behaviour, and the role of the number e. The topic also builds understanding of continuity and derivatives of exponential functions, helping students analyse change and solve related calculus problems accurately.
1 questionsLogarithmic functions
लघुगणकीय फलन
In this Class 12 Mathematics topic, students study logarithmic functions as inverses of exponential functions and understand their domain, range, base restrictions, graphs, and fundamental logarithm laws. Within the context of Continuity and Differentiability, they examine continuity and differentiability of logarithmic expressions, learn derivatives such as d/dx(log_a x) and d/dx(ln x), and apply these ideas to simplify functions, solve related problems, and analyse rates of change.
0 questionsLogarithmic differentiation
लघुगणकीय अवकलन
In this Class 12 Mathematics topic from Continuity and Differentiability, students learn how logarithms can simplify the differentiation of complicated expressions. The method is especially useful for products, quotients, powers, and functions in which both the base and exponent contain variables, such as y = u(x)^v(x). Students take logarithms, differentiate implicitly, apply derivative rules, and then express the result in terms of the original function. The topic also highlights domain and differentiability conditions, helping students solve derivatives accurately and efficiently.
0 questionsParametric differentiation
पैरामीट्रिक फलनों का अवकलन
In this Class 12 Mathematics topic from Continuity and Differentiability, students learn how to differentiate equations in which x and y are expressed in terms of a common parameter, usually t. They use the relation dy/dx = (dy/dt)/(dx/dt), understand the condition that dx/dt should be non-zero, and apply the method to find slopes, tangents, and higher-order derivatives of parametric curves. Examples help connect the procedure with geometric interpretation.
0 questionsSecond order derivatives
द्वितीय कोटि के अवकलज
In this Class 12 Mathematics topic from Continuity and Differentiability, students study second order derivatives, obtained by differentiating the first derivative again. They learn standard notation such as d²y/dx², calculate second derivatives for common functions, and apply differentiation rules accurately. The topic also develops understanding of how the second derivative describes the change in slope and helps analyse the concavity of a curve, along with its role in identifying local maxima and minima.
0 questions