Summary

Continuity and Differentiability Concepts

1. Continuity at an interior point

A function $f$ is continuous at $x=c$ if $$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c).$$ In words, the left-hand limit, right-hand limit, and actual function value must all exist and be equal.

2. Continuity on a closed interval

If the function is defined on $[a,b]$, then continuity at the endpoints is checked using one-sided limits: $$\lim_{x \to a^+} f(x)=f(a), \qquad \lim_{x \to b^-} f(x)=f(b).$$

3. Algebra of Continuous Functions

If $f$ and $g$ are continuous at $c$, then $f+g$, $f-g$, and $f \cdot g$ are continuous at $c$. Also, $$\frac{f}{g}$$ is continuous at $c$ provided $g(c) \neq 0$.

4. Composite Functions

If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$, then $$(f \circ g)(x)=f(g(x))$$ is continuous at $c$.

5. Differentiability at a point

A function $f$ is differentiable at $x=c$ if $$f'(c)=\lim_{h \to 0} \frac{f(c+h)-f(c)}{h}$$ exists as a finite real number.

6. LHD and RHD test for differentiability

For differentiability at $x=c$, we must have $$\lim_{h \to 0^-} \frac{f(c+h)-f(c)}{h} = \lim_{h \to 0^+} \frac{f(c+h)-f(c)}{h}.$$

7. Core theorem

Every differentiable function is continuous, but every continuous function need not be differentiable. The standard example is $$f(x)=|x|,$$ which is continuous at $x=0$ but not differentiable there because the slopes from the left and right are different.

8. Where non-differentiability usually occurs

sharp corners, cusps, vertical tangents, jump discontinuities, and oscillatory behavior near a point.

Rules of Differentiation

Let $u=u(x)$ and $v=v(x)$ be differentiable functions.

1. Sum and Difference Rule

$$\frac{d}{dx}(u \pm v)=\frac{du}{dx} \pm \frac{dv}{dx}$$

2. Product Rule (Leibnitz Rule)

$$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$$

3. Quotient Rule

$$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}, \qquad v \neq 0$$

4. Chain Rule

If $y=f(u)$ and $u=g(x)$, then $$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}.$$ This is the backbone of almost every advanced derivative question.

5. Repeated Chain Rule

For a composition like $y=f(g(h(x)))$, $$\frac{dy}{dx}=f'(g(h(x)))\cdot g'(h(x))\cdot h'(x).$$ Important reminder: In questions involving inverse trigonometric, exponential, logarithmic, or parametric forms, the chain rule is almost always used somewhere, even if it is hidden.

Master List of Standard Derivatives

1.Algebraic & Trigonometric

• $$\frac{d}{dx}(x^n)=n x^{n-1}$$
• $$\frac{d}{dx}(\sin x)=\cos x$$
• $$\frac{d}{dx}(\cos x)=-\sin x$$
• $$\frac{d}{dx}(\tan x)=\sec^2 x$$
• $$\frac{d}{dx}(\cot x)=-\csc^2 x$$
• $$\frac{d}{dx}(\sec x)=\sec x\tan x$$
• $$\frac{d}{dx}(\csc x)=-\csc x\cot x$$

2. Inverse Trigonometric

• $$\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}}, \quad -1<x<1$$
• $$\frac{d}{dx}(\cos^{-1}x)=\frac{-1}{\sqrt{1-x^2}}, \quad -1<x<1$$
• $$\frac{d}{dx}(\tan^{-1}x)=\frac{1}{1+x^2}$$
• $$\frac{d}{dx}(\cot^{-1}x)=\frac{-1}{1+x^2}$$
• $$\frac{d}{dx}(\sec^{-1}x)=\frac{1}{|x|\sqrt{x^2-1}}, \quad |x|>1$$
• $$\frac{d}{dx}(\csc^{-1}x)=\frac{-1}{|x|\sqrt{x^2-1}}, \quad |x|>1$$

3. Exponential & Logarithmic

• $$\frac{d}{dx}(e^x)=e^x$$
• $$\frac{d}{dx}(a^x)=a^x\ln a, \quad a>0, a\neq 1$$
• $$\frac{d}{dx}(\ln x)=\frac{1}{x}, \quad x>0$$
• $$\frac{d}{dx}(\log_a x)=\frac{1}{x\ln a}, \quad a>0, a\neq 1$$

4. Useful composite forms to remember

• $$\frac{d}{dx}(e^{u})=e^{u}\cdot u'$$
• $$\frac{d}{dx}(\ln u)=\frac{u'}{u}$$
• $$\frac{d}{dx}(a^{u})=a^{u}\ln a \cdot u'$$
• $$\frac{d}{dx}(\sin^{-1}u)=\frac{u'}{\sqrt{1-u^2}}$$
• $$\frac{d}{dx}(\tan^{-1}u)=\frac{u'}{1+u^2}$$

Advanced Differentiation Techniques

1. Implicit Differentiation

If a relation is given as $$F(x,y)=0,$$ then differentiate both sides with respect to $x$. Every time $y$ is differentiated, multiply by $\frac{dy}{dx}$ due to the chain rule. For example, $$\frac{d}{dx}(y^2)=2y\frac{dy}{dx}, \qquad \frac{d}{dx}(\sin y)=\cos y\frac{dy}{dx}.$$ After differentiation, collect all $\frac{dy}{dx}$ terms on one side and solve.

2. Logarithmic Properties

$$\ln(xy)=\ln x+\ln y,$$ $$\ln\left(\frac{x}{y}\right)=\ln x-\ln y,$$ $$\ln(x^n)=n\ln x,$$ $$\log_a x = \frac{\ln x}{\ln a}.$$

3. Logarithmic Differentiation

This is especially useful when the function is of the form $$y=u(x)^{v(x)}$$ or is a complicated product/quotient with powers. The method is:

1. Take natural logarithm on both sides.
2. Simplify using logarithmic laws.
3. Differentiate implicitly.
4. Replace $y$ by the original expression in the final answer.

Important warning: $$\ln(u+v) \neq \ln u + \ln v.$$ So logarithmic differentiation cannot be applied directly to a sum.

4. Parametric Differentiation

If $$x=f(t), \qquad y=g(t),$$ then $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}, \qquad \frac{dx}{dt} \neq 0.$$

5. Second Order Derivative in Parametric Form

$$\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot \frac{dt}{dx}.$$ This is the correct formula. It is not equal to $\frac{d^2y/dt^2}{d^2x/dt^2}$.

6. Differentiation with respect to another function

If $u=u(x)$ and $v=v(x)$, then $$\frac{du}{dv}=\frac{du/dx}{dv/dx}, \qquad \frac{dv}{dx} \neq 0.$$

Exam Tips for Board Exams

Continuity proofs

Always write LHL, RHL, and function value separately. Then conclude continuity by showing $$LHL = RHL = f(c).$$

Differentiability proofs

Write the definitions of LHD and RHD clearly before evaluating them. This presentation is often rewarded in board marking schemes.

Implicit differentiation

Start with the sentence: Differentiating both sides with respect to $x$. Then show the chain rule properly for terms containing $y$.

Logarithmic differentiation

If the function is a sum like $x^x + (\sin x)^x$, do not take logarithm of the whole expression. Split it into separate parts first.

Second derivative proofs

When asked to prove expressions such as $$(1-x^2)y'' - xy' = 2,$$ it is often easier to first simplify or rearrange the equation involving $y'$ before differentiating again.

Substitute back in final answers

After logarithmic differentiation, never leave the result in terms of $y$ if the original question asked for derivative in terms of $x$.

Write domain restrictions when needed

For example, while differentiating $\ln x$, write that $x>0$; for inverse trigonometric functions, note their valid intervals when relevant.

Be careful with notation

$y'$, $\frac{dy}{dx}$, $y''$, and $\frac{d^2y}{dx^2}$ should be used neatly and consistently.

Exam Tips for JEE Main & Advanced

Spot non-differentiability quickly

For $|g(x)|$, likely trouble occurs where $g(x)=0$. If the sign changes there, the graph usually has a corner. For sums of modulus terms, check all such points.

Max/Min functions

For $$f(x)=\max(g(x),h(x)) \quad \text{or} \quad f(x)=\min(g(x),h(x)),$$ the likely critical points are where $g(x)=h(x)$.

Inverse trigonometric simplification

If expressions like $$\frac{2x}{1+x^2}, \quad \frac{1-x^2}{1+x^2}, \quad \frac{2x}{1-x^2}, \quad \frac{3x-x^3}{1-3x^2}$$ appear inside inverse trigonometric functions, look for substitutions such as $x=\tan\theta$ and use double-angle or triple-angle identities.

Parametric second derivative trap

Memorize the correct formula: $$\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot\frac{dt}{dx}.$$ Never replace it by a naive ratio of second derivatives with respect to $t$.

Infinite nested structures

If the expression repeats forever, assign the repeating block to a variable and convert the infinite form into an algebraic or implicit equation.

Oscillatory functions near zero

For functions containing $\sin(1/x)$ or $\cos(1/x)$, use squeeze theorem ideas for continuity and the first-principle derivative for differentiability.

Functional equations

When a question involves something like $$f(x+y)=f(x)+f(y)+xy$$ or $$f(x+y)=f(x)f(y),$$ substitute $y=h$ and use first principles.

Compositions

In repeated compositions like $f(f(f(x)))$, apply the chain rule layer by layer. Missing even one factor is a common JEE mistake.

Time-saving principle

Before differentiating a long expression directly, first try to simplify it using identities, factorization, substitution, or logarithms. In competitive exams, simplification is often the real key step.