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Last-Minute Formula Sheet — ITF, Differentiation, Continuity, AOD & Mensuration

Last-Minute Formula Sheet

Inverse Trigonometric Functions (ITF)

Principal-value domains and ranges

Core identities

\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}

\tan^{-1}x+\cot^{-1}x=\frac{\pi}{2}

\sec^{-1}x+\cosec^{-1}x=\frac{\pi}{2}

\sin^{-1}(-x)=-\sin^{-1}x,\quad \tan^{-1}(-x)=-\tan^{-1}x

\cos^{-1}(-x)=\pi-\cos^{-1}x,\quad \cot^{-1}(-x)=\pi-\cot^{-1}x

Addition and subtraction

\tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\frac{x+y}{1-xy}\right),\quad xy<1

\tan^{-1}x-\tan^{-1}y=\tan^{-1}\left(\frac{x-y}{1+xy}\right),\quad xy>-1

For \(xy>1\), adjust the result by \(\pm\pi\) according to the principal range.

2\tan^{-1}x=\tan^{-1}\left(\frac{2x}{1-x^2}\right)

Useful conversions:

\sin^{-1}x=\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right),\quad |x|<1

\cos^{-1}x=\tan^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)\quad (x>0)

\tan^{-1}x=\sin^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)

Differentiation — Important Formulas

Basic derivatives

\frac{d}{dx}(c)=0,\quad \frac{d}{dx}(x^n)=nx^{n-1}

\frac{d}{dx}(e^x)=e^x,\quad \frac{d}{dx}(a^x)=a^x\log a

\frac{d}{dx}(\log x)=\frac{1}{x},\quad \frac{d}{dx}(\log_a x)=\frac{1}{x\log a}

Trigonometric functions

(\sin x)'=\cos x,\quad (\cos x)'=-\sin x

(\tan x)'=\sec^2x,\quad (\cot x)'=-\cosec^2x

(\sec x)'=\sec x\tan x,\quad (\cosec x)'=-\cosec x\cot x

Inverse trigonometric functions

(\sin^{-1}x)'=\frac{1}{\sqrt{1-x^2}}

(\cos^{-1}x)'=-\frac{1}{\sqrt{1-x^2}}

(\tan^{-1}x)'=\frac{1}{1+x^2}

(\cot^{-1}x)'=-\frac{1}{1+x^2}

(\sec^{-1}x)'=\frac{1}{|x|\sqrt{x^2-1}}

(\cosec^{-1}x)'=-\frac{1}{|x|\sqrt{x^2-1}}

Rules

(u\pm v)'=u'\pm v'

(uv)'=u'v+uv'

\left(\frac{u}{v}\right)'=\frac{vu'-uv'}{v^2}

\frac{d}{dx}f(g(x))=f'(g(x))g'(x)

Chain-rule forms

(\sin u)'=\cos u\cdot u',\quad (\cos u)'=-\sin u\cdot u'

(\tan u)'=\sec^2u\cdot u',\quad (e^u)'=e^u\cdot u'

(\log u)'=\frac{u'}{u}

(\sin^{-1}u)'=\frac{u'}{\sqrt{1-u^2}},\quad (\tan^{-1}u)'=\frac{u'}{1+u^2}

Parametric and implicit differentiation

If \(x=f(t)\), \(y=g(t)\), then

\frac{dy}{dx}=\frac{dy/dt}{dx/dt}

If \(F(x,y)=0\), differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\).

Logarithmic differentiation

If \(y=u^v\), then

\log y=v\log u

\frac{y'}{y}=v'\log u+v\frac{u'}{u}

Continuity and Differentiability

Continuity at \(x=a\)

\lim_{x\to a}f(x)=f(a)

Equivalent condition:

\text{LHL}=\text{RHL}=f(a)

For a piecewise function, calculate the left-hand limit, right-hand limit, and actual value separately.

Differentiability at \(x=a\)

\text{LHD}=\text{RHD}

where

\text{LHD}=\lim_{h\to0^-}\frac{f(a+h)-f(a)}{h},

\quad

\text{RHD}=\lim_{h\to0^+}\frac{f(a+h)-f(a)}{h}

Key facts

Differentiability implies continuity.

Continuity does not necessarily imply differentiability.

A function may fail to be differentiable at a corner, cusp, sharp point, or discontinuity.

Polynomials, exponential functions, logarithmic functions on their domains, and standard trigonometric functions are continuous and differentiable wherever defined.

\(|x|\) is continuous at \(x=0\), but not differentiable there.

Common limits

\lim_{x\to0}\frac{\sin x}{x}=1

\lim_{x\to0}\frac{\tan x}{x}=1

\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12

\lim_{x\to0}\frac{e^x-1}{x}=1

\lim_{x\to0}\frac{\log(1+x)}{x}=1

Applications of Derivatives — Increasing and Decreasing

For a function \(f(x)\):

If \(f'(x)>0\) on an interval, \(f\) is increasing there.

If \(f'(x)<0\) on an interval, \(f\) is decreasing there.

If \(f'(x)=0\), the point is a critical/stationary point; test the sign of \(f'\) on both sides.

Sign-chart method

Find \(f'(x)\).

Solve \(f'(x)=0\) and identify points where \(f'(x)\) is undefined.

Mark these points on the number line.

Test the sign of \(f'(x)\) in each interval.

Write intervals of increase and decrease.

First-derivative test

f':+\to- \Rightarrow \text{local maximum}

f':-\to+ \Rightarrow \text{local minimum}

For \(f'(x)=0\) throughout an interval, the function is constant on that interval.

Mensuration — Cube and Cuboid

Cuboid

Let length \(l\), breadth \(b\), height \(h\).

\text{Volume}=lbh

\text{LSA}=2h(l+b)

\text{TSA}=2(lb+bh+hl)

\text{Diagonal}=\sqrt{l^2+b^2+h^2}

Cube

Let side \(a\).

\text{Volume}=a^3

\text{LSA}=4a^2

\text{TSA}=6a^2

\text{Face diagonal}=a\sqrt2

\text{Space diagonal}=a\sqrt3

Mensuration — Cylinder

Let radius \(r\), height \(h\).

\text{Base area}=\pi r^2

\text{CSA}=2\pi rh

\text{TSA}=2\pi r(h+r)

\text{Volume}=\pi r^2h

For a hollow cylinder with outer radius \(R\), inner radius \(r\), height \(h\):

\text{Volume of material}=\pi h(R^2-r^2)

Mensuration — Cone

Let radius \(r\), height \(h\), slant height \(l\).

l=\sqrt{r^2+h^2}

\text{CSA}=\pi rl

\text{TSA}=\pi r(l+r)

\text{Volume}=\frac13\pi r^2h

Frustum of a cone

Let larger radius \(R\), smaller radius \(r\), height \(h\), slant height \(l\).

l=\sqrt{h^2+(R-r)^2}

\text{CSA}=\pi(R+r)l

\text{TSA}=\pi(R+r)l+\pi R^2+\pi r^2

\text{Volume}=\frac13\pi h(R^2+r^2+Rr)

Mensuration — Sphere and Hemisphere

Sphere

Let radius \(r\).

\text{Surface area}=4\pi r^2

\text{Volume}=\frac43\pi r^3

Hemisphere

\text{CSA}=2\pi r^2

\text{TSA}=3\pi r^2

\text{Volume}=\frac23\pi r^3

Quick reminders

Use the same units throughout.

Area is in square units; volume is in cubic units.

\(\text{CSA/LSA}\) excludes the base(s); \(\text{TSA}\) includes all exposed surfaces.

In calculus questions, use radians for trigonometric differentiation and limits.

For ITF answers, always check the principal range before finalising the value.