Differentiation — JEE Mains Maths Formulas and Traps

Chain Rule and Inverse-Trig Simplification

Simplify before you differentiate

A standard substitution

x=\tan\theta:\quad\tan^{-1}\frac{2x}{1-x^2}=2\theta=2\tan^{-1}x\quad(|x|<1)

The chain rule and inverse functions

Derivative of an inverse

g'(k)=\frac{1}{f'(a)},\qquad f(a)=k

Common traps

The branch decides the answer

\cos^{-1}(\cos x)=x

only on

[0,\pi]

. On

[\pi,2\pi]

it is

2\pi-x

, whose derivative is

-1

, not

1

. Read the stated interval before you simplify.

Invert at the right point

g'(k)

is

\frac{1}{f'(a)}

where

f(a)=k

, not

\frac{1}{f'(k)}

. Solve

f(a)=k

first; the root is usually a small integer.

Implicit, Parametric and Logarithmic Differentiation

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Implicit and parametric differentiation

Second derivative, parametric

\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\Big/\frac{dx}{dt}

Logarithmic differentiation

Variable power

\frac{d}{dx}f^{\,g}=f^{\,g}\left(g'\ln f+\frac{g\,f'}{f}\right)

Common traps

Not the ratio of second derivatives

For a parametric curve

\frac{d^2y}{dx^2}\ne\frac{\ddot y}{\ddot x}

. With

x=t^2,\ y=t^3

the ratio gives

3t

, but the true value is

\frac{3}{4t}

. Differentiate

\frac{dy}{dx}

with respect to

t

, then divide by

\frac{dx}{dt}

.

d²x/dy² is not 1/y″

\frac{dx}{dy}=\frac{1}{y'}

, but

\frac{d^2x}{dy^2}=-\frac{y''}{(y')^3}

, not

\frac{1}{y''}

. Differentiate

\frac{1}{y'}

with respect to

x

, then divide by

y'

once more.

Functional Equations and Derivative Constants

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Functional equations

Derivative from the equation

f(x+y)=f(x)f(y)\ \Rightarrow\ f'(x)=\lim_{h\to0}f(x)\frac{f(h)-1}{h}=f'(0)\,f(x)

Derivative values as constants

For a cubic

f(x)=x^3+ax^2+bx+c\ \Rightarrow\ f'(x)=3x^2+2ax+b,\quad f''(x)=6x+2a,\quad f'''(x)=6

Common traps

Find f(0) first

Most of these equations fix

f(0)

before anything else:

f(0)=f(0)^2

gives

f(0)=1

when

f

never vanishes, and

f(0)=2f(0)

gives

f(0)=0

. Skipping it leaves an unknown constant in the answer.

The coefficient is not a function

f'(1)

is a number, so

x^2f'(1)

differentiates to

2xf'(1)

. Treating

f'(1)

as a function of

x

and using the product rule gives wrong equations.

Differentiability of Piecewise Functions

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Where two pieces join

Smooth join at x = a

f_1(a)=f_2(a)\quad\text{and}\quad f_1'(a)=f_2'(a)

Composites and running maxima

Chain rule at a point

(g\circ f)'(a)=g'\big(f(a)\big)\,f'(a)\quad\text{when both derivatives exist}

The derivative from its definition

Derivative at a point

f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}

Common traps

Continuity comes first

Matching slopes is not enough.

x

for

x<0

and

x+1

for

x\ge0

have slope 1 on both sides but jump at 0, so the function is not differentiable there. Write both equations.

A bad inner function can be smoothed out

|x|

has a corner at 0, but

|x|^2=x^2

is smooth. Test

g\circ f

itself at each suspect point; do not assume it fails because

f

does.

f′ can exist without being continuous

For

x^2\sin\frac1x

, the formula for

f'(x)

has no limit at 0, but

f'(0)=0

from the definition. Never decide differentiability at a point by taking the limit of the derivative formula.

Counting Points of Non-Differentiability

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Corners of modulus, max and min

Slope jump at a corner

k\,|x-a|:\quad f'(a^+)-f'(a^-)=2k

Jumps of the greatest integer function

Greatest integer function

[x]=n\ \text{ for }\ n\le x<n+1,\qquad[x+n]=[x]+n\ \ (n\in\mathbb{Z})

Common traps

A candidate is not yet a corner

Every zero inside a modulus is only a candidate.

|x-1|\sin|x-1|

and

e^{|(x-1)^2|}

are smooth at 1, and a factor that vanishes at the same point removes the corner. Test each candidate before counting it.

A jump counts twice in m + n

When a question asks for the points where

f

is not continuous (

m

) and not differentiable (

n

), a jump belongs to both counts. Add every jump to

n

as well as to

m

.

Differentiation formulas

Chain Rule and Inverse-Trig Simplification

Simplify before you differentiate

The chain rule and inverse functions

Common traps

Implicit, Parametric and Logarithmic Differentiation

Implicit and parametric differentiation

Logarithmic differentiation

Common traps

Functional Equations and Derivative Constants

Functional equations

Derivative values as constants

Common traps

Differentiability of Piecewise Functions

Where two pieces join

Composites and running maxima

The derivative from its definition

Common traps

Counting Points of Non-Differentiability

Corners of modulus, max and min

Jumps of the greatest integer function

Common traps

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