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Method OF Differentiation formulas

Master Method OF Differentiation through 34 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Method OF Differentiation, every formula

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Derivative (definition)
$f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}$
Q1MCQDefinition
$f'(x)$ is defined as:
- A
  $\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$
- B
  $\lim_{h\to0}f(x+h)$
- C
  $f(x)$
- D
  $\lim_{x\to0}f(x)$
Constant
$\frac{d}{d x} (c) = 0$
Power rule
$\frac{d}{d x} (x^{n}) = n x^{n - 1}$
Q1MCQPower rule
$\dfrac{d}{dx}(x^{5})$ equals:
- A
  $5x^{4}$
- B
  $x^{4}$
- C
  $5x^{5}$
- D
  $4x^{5}$
Exponential
$\frac{d}{d x} (e^{x}) = e^{x}$
Q1MCQExponential
$\dfrac{d}{dx}(e^{x})$ equals:
- A
  $e^{x}$
- B
  $xe^{x-1}$
- C
  $e^{x}\ln x$
- D
  $\dfrac{e^{x}}{x}$
General exponential
$\frac{d}{d x} (a^{x}) = a^{x} ln a$
Q1MCQGeneral exponential
$\dfrac{d}{dx}(2^{x})$ equals:
- A
  $2^{x}\ln 2$
- B
  $x2^{x-1}$
- C
  $2^{x}$
- D
  $\dfrac{2^{x}}{\ln 2}$
Logarithm
$\frac{d}{d x} (ln ∣ x ∣) = \frac{1}{x}$
Q1MCQLogarithm
$\dfrac{d}{dx}(\ln x)$ equals:
- A
  $\dfrac1x$
- B
  $\ln x$
- C
  $x$
- D
  $-\dfrac{1}{x^{2}}$
Sum/difference rule
$(c_{1} f \pm c_{2} g)^{'} = c_{1} f^{'} \pm c_{2} g^{'}$
Q1MCQSum rule
$\dfrac{d}{dx}(3x^{2}+2x)$ equals:
- A
  $6x+2$
- B
  $3x+2$
- C
  $6x$
- D
  $5x$
Product rule
$(f g)^{'} = f^{'} g + f g^{'}$
Q1MCQProduct rule
$\dfrac{d}{dx}(x^{2}e^{x})$ equals:
- A
  $e^{x}(x^{2}+2x)$
- B
  $2xe^{x}$
- C
  $x^{2}e^{x}$
- D
  $e^{x}(x^{2}-2x)$
Quotient rule
$(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}$
Q1MCQQuotient rule
$\dfrac{d}{dx}\dfrac{x}{x+1}$ equals:
- A
  $\dfrac{1}{(x+1)^{2}}$
- B
  $\dfrac{-1}{(x+1)^{2}}$
- C
  $\dfrac{x}{(x+1)^{2}}$
- D
  $1$
Chain rule
$\frac{d y}{d x} = \frac{d y}{d t} \cdot \frac{d t}{d x}$
Q1MCQChain rule
$\dfrac{d}{dx}\sin(x^{2})$ equals:
- A
  $2x\cos(x^{2})$
- B
  $\cos(x^{2})$
- C
  $2x\sin(x^{2})$
- D
  $\cos(2x)$
d/dx sin x
$\frac{d}{d x} (sin x) = cos x$
Q1MCQsin x
$\dfrac{d}{dx}(\sin x)$ equals:
- A
  $\cos x$
- B
  $-\cos x$
- C
  $-\sin x$
- D
  $\sec^{2}x$
d/dx cos x
$\frac{d}{d x} (cos x) = - sin x$
Q1MCQcos x
$\dfrac{d}{dx}(\cos x)$ equals:
- A
  $-\sin x$
- B
  $\sin x$
- C
  $-\cos x$
- D
  $\sec^{2}x$
d/dx tan x
$\frac{d}{d x} (tan x) = sec^{2} x$
Q1MCQtan x
$\dfrac{d}{dx}(\tan x)$ equals:
- A
  $\sec^{2}x$
- B
  $-\csc^{2}x$
- C
  $\sec x\tan x$
- D
  $\cos x$
d/dx cot x
$\frac{d}{d x} (cot x) = - csc^{2} x$
Q1MCQcot x
$\dfrac{d}{dx}(\cot x)$ equals:
- A
  $-\csc^{2}x$
- B
  $\sec^{2}x$
- C
  $\csc^{2}x$
- D
  $-\sec x\tan x$
d/dx sec x
$\frac{d}{d x} (sec x) = sec x tan x$
Q1MCQsec x
$\dfrac{d}{dx}(\sec x)$ equals:
- A
  $\sec x\tan x$
- B
  $-\csc x\cot x$
- C
  $\sec^{2}x$
- D
  $\tan x$
d/dx cosec x
$\frac{d}{d x} (csc x) = - csc x cot x$
Q1MCQcosec x
$\dfrac{d}{dx}(\csc x)$ equals:
- A
  $-\csc x\cot x$
- B
  $\sec x\tan x$
- C
  $-\csc^{2}x$
- D
  $\cot x$
d/dx arcsin
$\frac{d}{d x} (sin^{- 1} x) = \frac{1}{1 - x ^{2}}$
Q1MCQarcsin
$\dfrac{d}{dx}(\sin^{-1}x)$ equals:
- A
  $\dfrac{1}{\sqrt{1-x^{2}}}$
- B
  $\dfrac{-1}{\sqrt{1-x^{2}}}$
- C
  $\dfrac{1}{1+x^{2}}$
- D
  $\dfrac{1}{|x|\sqrt{x^{2}-1}}$
d/dx arccos
$\frac{d}{d x} (cos^{- 1} x) = \frac{- 1}{1 - x ^{2}}$
Q1MCQarccos
$\dfrac{d}{dx}(\cos^{-1}x)$ equals:
- A
  $\dfrac{-1}{\sqrt{1-x^{2}}}$
- B
  $\dfrac{1}{\sqrt{1-x^{2}}}$
- C
  $\dfrac{-1}{1+x^{2}}$
- D
  $\dfrac{1}{1+x^{2}}$
d/dx arctan
$\frac{d}{d x} (tan^{- 1} x) = \frac{1}{1 + x ^{2}}$
Q1MCQarctan
$\dfrac{d}{dx}(\tan^{-1}x)$ equals:
- A
  $\dfrac{1}{1+x^{2}}$
- B
  $\dfrac{-1}{1+x^{2}}$
- C
  $\dfrac{1}{\sqrt{1-x^{2}}}$
- D
  $\dfrac{1}{1-x^{2}}$
d/dx arccot
$\frac{d}{d x} (cot^{- 1} x) = \frac{- 1}{1 + x ^{2}}$
Q1MCQarccot
$\dfrac{d}{dx}(\cot^{-1}x)$ equals:
- A
  $\dfrac{-1}{1+x^{2}}$
- B
  $\dfrac{1}{1+x^{2}}$
- C
  $\dfrac{-1}{\sqrt{1-x^{2}}}$
- D
  $\dfrac{1}{1-x^{2}}$
d/dx arcsec
$\frac{d}{d x} (sec^{- 1} x) = \frac{1}{∣ x ∣ x ^{2} - 1}$
Q1MCQarcsec
$\dfrac{d}{dx}(\sec^{-1}x)$ equals:
- A
  $\dfrac{1}{|x|\sqrt{x^{2}-1}}$
- B
  $\dfrac{1}{\sqrt{1-x^{2}}}$
- C
  $\dfrac{1}{1+x^{2}}$
- D
  $\dfrac{-1}{|x|\sqrt{x^{2}-1}}$
Log-derivative rule
$\frac{d}{d x} ln f (x) = \frac{f ^{'} ( x )}{f ( x )}$
Q1MCQLog-derivative
$\dfrac{d}{dx}\ln(x^{2}+1)$ equals:
- A
  $\dfrac{2x}{x^{2}+1}$
- B
  $\dfrac{1}{x^{2}+1}$
- C
  $2x$
- D
  $\dfrac{2x}{x^{2}}$
Root of a function
$\frac{d}{d x} f (x) = \frac{f ^{'} ( x )}{2 f ( x )}$
Q1MCQRoot of function
$\dfrac{d}{dx}\sqrt{x^{2}+1}$ equals:
- A
  $\dfrac{x}{\sqrt{x^{2}+1}}$
- B
  $\dfrac{2x}{\sqrt{x^{2}+1}}$
- C
  $\dfrac{1}{2\sqrt{x^{2}+1}}$
- D
  $\sqrt{x^{2}+1}$
Power of a function
$\frac{d}{d x} [f (x)]^{n} = n [f (x)]^{n - 1} f^{'} (x)$
Q1MCQPower of function
$\dfrac{d}{dx}(x^{2}+1)^{3}$ equals:
- A
  $6x(x^{2}+1)^{2}$
- B
  $3(x^{2}+1)^{2}$
- C
  $(x^{2}+1)^{2}$
- D
  $6x(x^{2}+1)^{3}$
Parametric derivative
$\frac{d y}{d x} = \frac{g ^{'} ( t )}{f ^{'} ( t )}$
x=f(t), y=g(t)
Q1MCQParametric
For $x=t^{2},\ y=t^{3}$, $\dfrac{dy}{dx}$ equals:
- A
  $\dfrac{3t}{2}$
- B
  $\dfrac{2}{3t}$
- C
  $3t^{2}$
- D
  $\dfrac{2t}{3}$
Implicit differentiation
$\frac{d y}{d x} = - \frac{\partial f / \partial x}{\partial f / \partial y}$
f(x,y)=0
Q1MCQImplicit
For $x^{2}+y^{2}=25$, $\dfrac{dy}{dx}$ equals:
- A
  $-\dfrac{x}{y}$
- B
  $\dfrac{x}{y}$
- C
  $-\dfrac{y}{x}$
- D
  $\dfrac{y}{x}$
Logarithmic differentiation
$y = f^{g} : ln y = g ln f, then differentiate$
Q1MCQLog differentiation
For $y=x^{x}$, we first take:
- A
  $\ln y=x\ln x$
- B
  $\ln y=x$
- C
  $y=e^{x}$
- D
  $\ln y=\ln x$
Derivative of f(x)^{g(x)}
$y^{'} = y [g^{'} ln f + g \frac{f ^{'}}{f}]$
Q1MCQf^g derivative
For $y=f(x)^{g(x)}$, $y'$ equals:
- A
  $y\left[g'\ln f+g\dfrac{f'}{f}\right]$
- B
  $gf^{g-1}f'$
- C
  $f^{g}\ln f$
- D
  $g'f^{g}$
Higher-order derivative
$\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{d y}{d x})$
d/dx (ax+b)/(cx+d)
$\frac{d}{d x} \frac{a x + b}{c x + d} = \frac{a d - b c}{( c x + d ) ^{2}}$
Q1MCQ(ax+b)/(cx+d)
$\dfrac{d}{dx}\dfrac{2x+1}{x+1}$ equals:
- A
  $\dfrac{1}{(x+1)^{2}}$
- B
  $\dfrac{2}{(x+1)^{2}}$
- C
  $\dfrac{3}{(x+1)^{2}}$
- D
  $\dfrac{-1}{(x+1)^{2}}$
Derivative of even is odd
$f even \Rightarrow f^{'} odd (and f^{'} (0) = 0)$
√(a²−x²) substitution
$x = a sin θ$
Q1MCQ√(a²−x²) sub
To differentiate a function with $\sqrt{a^{2}-x^{2}}$, substitute:
- A
  $x=a\sin\theta$
- B
  $x=a\tan\theta$
- C
  $x=a\sec\theta$
- D
  $x=a\cos 2\theta$
√(a²+x²) substitution
$x = a tan θ$
Q1MCQ√(a²+x²) sub
To differentiate a function with $\sqrt{a^{2}+x^{2}}$, substitute:
- A
  $x=a\tan\theta$
- B
  $x=a\sin\theta$
- C
  $x=a\sec\theta$
- D
  $x=a\cos\theta$
Infinite nested radical
$y = f (x) + f (x) + \dots \Rightarrow \frac{d y}{d x} = \frac{f ^{'} ( x )}{2 y - 1}$
Q1MCQNested radical
For $y=\sqrt{f(x)+\sqrt{f(x)+\cdots}}$ (to $\infty$), $\dfrac{dy}{dx}$ equals:
- A
  $\dfrac{f'(x)}{2y-1}$
- B
  $\dfrac{f'(x)}{2y}$
- C
  $\dfrac{f'(x)}{y}$
- D
  $2yf'(x)$

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Method OF Differentiation formulas

Method OF Differentiation, every formula

Derivative (definition)

Constant

Power rule

Exponential

General exponential

Logarithm

Sum/difference rule

Product rule

Quotient rule

Chain rule

d/dx sin x

d/dx cos x

d/dx tan x

d/dx cot x

d/dx sec x

d/dx cosec x

d/dx arcsin

d/dx arccos

d/dx arctan

d/dx arccot

d/dx arcsec

Log-derivative rule

Root of a function

Power of a function

Parametric derivative

Implicit differentiation

Logarithmic differentiation

Derivative of f(x)^{g(x)}

Higher-order derivative

d/dx (ax+b)/(cx+d)

Derivative of even is odd

√(a²−x²) substitution

√(a²+x²) substitution

Infinite nested radical

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