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Differentiability formulas

Master Differentiability through 26 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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Differentiability, every formula

26 formulas, typeset and free. Print it, or keep it open beside your practice.

Derivative at a point
$f^{'} (a) = x \to a lim \frac{f ( x ) - f ( a )}{x - a}$
if it exists finitely
Q1MCQDerivative at point
$f'(a)$ is defined as:
- A
  $\lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}$
- B
  $\lim_{x\to a}f(x)$
- C
  $f(a)$
- D
  $\lim_{x\to a}[f(x)-f(a)]$
Right-hand derivative
$f^{'} (a^{+}) = h \to 0 lim \frac{f ( a + h ) - f ( a )}{h}$
Q1MCQRHD
The right-hand derivative uses:
- A
  $\lim_{h\to0}\dfrac{f(a+h)-f(a)}{h}$
- B
  $\lim_{h\to0}\dfrac{f(a-h)-f(a)}{-h}$
- C
  $f(a+h)$
- D
  $\lim_{h\to0}f(a+h)$
Left-hand derivative
$f^{'} (a^{-}) = h \to 0 lim \frac{f ( a - h ) - f ( a )}{- h}$
Q1MCQLHD
The left-hand derivative uses:
- A
  $\lim_{h\to0}\dfrac{f(a-h)-f(a)}{-h}$
- B
  $\lim_{h\to0}\dfrac{f(a+h)-f(a)}{h}$
- C
  $f(a-h)$
- D
  $\lim_{h\to0}f(a-h)$
Differentiability condition
$f^{'} (a^{+}) = f^{'} (a^{-}) (both finite)$
Q1MCQDiff condition
$f$ is differentiable at $a$ iff:
- A
  $f'(a^{+})=f'(a^{-})$ (both finite)
- B
  $f(a)$ exists
- C
  $f$ is continuous
- D
  $\lim f$ exists
Differentiable ⇒ continuous
$f differentiable at a \Rightarrow f continuous at a$
converse false
Q1MCQDiff ⇒ cts
If $f$ is differentiable at $a$, then $f$ is:
- A
  continuous at $a$
- B
  discontinuous at $a$
- C
  constant
- D
  undefined
Discontinuous ⇒ non-differentiable
$f discontinuous at a \Rightarrow f not differentiable at a$
Q1MCQDiscts ⇒ non-diff
If $f$ is discontinuous at $a$, then $f$ is:
- A
  not differentiable at $a$
- B
  differentiable at $a$
- C
  bounded
- D
  constant
Differentiable on (a,b)
$differentiable at each point of (a, b)$
Q1MCQOn (a,b)
$f$ is differentiable on $(a,b)$ if it is differentiable:
- A
  at each point of $(a,b)$
- B
  only at the midpoint
- C
  at the endpoints
- D
  nowhere
Differentiable on [a,b]
$diff. on (a, b), right-deriv at a, left-deriv at b$
Q1MCQOn [a,b]
Differentiability on $[a,b]$ additionally needs, at the endpoints:
- A
  one-sided derivatives
- B
  $f(a)=f(b)$
- C
  continuity only
- D
  nothing
Sum of differentiable
$f \pm g differentiable$
if f,g differentiable
Q1MCQSum diff
If $f,g$ are differentiable, then $f+g$ is:
- A
  differentiable
- B
  non-differentiable
- C
  discontinuous
- D
  constant
Product of differentiable
$f \cdot g differentiable$
Q1MCQProduct diff
If $f,g$ are differentiable, then $fg$ is:
- A
  differentiable
- B
  non-differentiable
- C
  $0$
- D
  discontinuous
Diff ± non-diff
$differentiable \pm non-differentiable = non-differentiable$
Q1MCQDiff ± non-diff
(differentiable) $+$ (non-differentiable) is:
- A
  non-differentiable
- B
  differentiable
- C
  continuous
- D
  constant
Polynomials differentiable
$polynomials, e^{x}, sin, cos, lo g differentiable on their domains$
Q1MCQPolynomials
Which is differentiable everywhere on $\mathbb{R}$?
- A
  $f(x)=x^{3}+e^{x}$
- B
  $f(x)=|x|$
- C
  $f(x)=[x]$
- D
  $f(x)=\{x\}$
|x-a| non-differentiable
$∣ x - a ∣ is not differentiable at x = a$
sharp corner
Q1MCQ|x-a|
$f(x)=|x-2|$ is not differentiable at:
- A
  $x=2$
- B
  $x=0$
- C
  $x=-2$
- D
  everywhere
(x-a)^n|x-a|
$(x - a)^{n} ∣ x - a ∣ differentiable at a iff n \geq 1$
Q1MCQ(x-a)^n|x-a|
$(x-2)^{2}|x-2|$ at $x=2$ is:
- A
  differentiable
- B
  non-differentiable
- C
  discontinuous
- D
  undefined
sgn(x-a) non-differentiable
$sgn (x - a) not differentiable at x = a$
jump
Q1MCQsgn
$\operatorname{sgn}(x-3)$ at $x=3$ is:
- A
  non-differentiable
- B
  differentiable
- C
  continuous
- D
  constant
xⁿ sin(1/x) differentiability
$x^{n} sin \frac{1}{x} differentiable at 0 iff n > 1$
Q1MCQxⁿ sin(1/x)
$x^{2}\sin\dfrac1x$ (with $f(0)=0$) at $x=0$ is:
- A
  differentiable
- B
  non-differentiable
- C
  discontinuous
- D
  undefined
{x},[x] at integers
${x}, [x] not differentiable at integers$
Q1MCQ{x},[x]
$[x]$ (greatest integer) is not differentiable at:
- A
  all integers
- B
  all reals
- C
  $x=0.5$
- D
  nowhere
Leibnitz differentiation
$\frac{d}{d x} \int_{α (x)}^{β (x)} f (t) d t = f (β) β^{'} - f (α) α^{'}$
Q1MCQLeibnitz
$\dfrac{d}{dx}\displaystyle\int_{0}^{x^{2}}f(t)\,dt$ equals:
- A
  $f(x^{2})\cdot 2x$
- B
  $f(x^{2})$
- C
  $2x$
- D
  $f(x)$
Vertical tangent
$f continuous, ∣ f^{'} (x) ∣ \to \infty at a \Rightarrow vertical tangent, non-diff.$
Q1MCQVertical tangent
A continuous function with $|f'(x)|\to\infty$ at $a$ has:
- A
  a vertical tangent (non-differentiable)
- B
  a horizontal tangent
- C
  a corner
- D
  a jump
Oscillation point
$high-frequency oscillation \Rightarrow non-differentiable$
Q1MCQOscillation point
A continuous function whose one-sided derivatives fail due to rapid oscillation has:
- A
  an oscillation point (non-differentiable)
- B
  a corner
- C
  a vertical tangent
- D
  a jump
FE f(x+y)=f(x)f(y)
$f (x + y) = f (x) f (y) \Rightarrow f^{'} (x) = f^{'} (0) f (x)$
Q1MCQFE product
If $f(x+y)=f(x)f(y)$, then $f'(x)$ equals:
- A
  $f'(0)f(x)$
- B
  $f(x)$
- C
  $f'(0)$
- D
  $0$
FE f(x+y)=f(x)+f(y)
$f (x + y) = f (x) + f (y) \Rightarrow f (x) = k x$
Q1MCQFE additive
If $f(x+y)=f(x)+f(y)$ for all $x,y$, then $f(x)$ is:
- A
  $kx$
- B
  $k^{x}$
- C
  $k\log x$
- D
  $1\pm x^{n}$
FE f(xy)=f(x)+f(y)
$f (x y) = f (x) + f (y) \Rightarrow f (x) = k lo g x$
Q1MCQFE log
If $f(xy)=f(x)+f(y)$, then $f(x)$ is:
- A
  $k\log x$
- B
  $kx$
- C
  $k^{x}$
- D
  $1\pm x^{n}$
FE f(x)f(1/x)=f(x)+f(1/x)
$f (x) f (\frac{1}{x}) = f (x) + f (\frac{1}{x}) \Rightarrow f (x) = 1 \pm x^{n}$
Q1MCQFE poly
If $f(x)f\!\left(\tfrac1x\right)=f(x)+f\!\left(\tfrac1x\right)$, then $f(x)$ is:
- A
  $1\pm x^{n}$
- B
  $kx$
- C
  $k^{x}$
- D
  $k\log x$
Corner ⇒ non-differentiable
$sharp corner (LHD \neq = RHD) \Rightarrow non-differentiable$
Q1MCQCorner
At a sharp corner where LHD $\neq$ RHD, the function is:
- A
  non-differentiable
- B
  differentiable
- C
  discontinuous
- D
  constant
Constant function derivative
$\frac{d}{d x} (c) = 0$

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Differentiability formulas

Differentiability, every formula

Derivative at a point

Right-hand derivative

Left-hand derivative

Differentiability condition

Differentiable ⇒ continuous

Discontinuous ⇒ non-differentiable

Differentiable on (a,b)

Differentiable on [a,b]

Sum of differentiable

Product of differentiable

Diff ± non-diff

Polynomials differentiable

|x-a| non-differentiable

(x-a)^n|x-a|

sgn(x-a) non-differentiable

xⁿ sin(1/x) differentiability

{x},[x] at integers

Leibnitz differentiation

Vertical tangent

Oscillation point

FE f(x+y)=f(x)f(y)

FE f(x+y)=f(x)+f(y)

FE f(xy)=f(x)+f(y)

FE f(x)f(1/x)=f(x)+f(1/x)

Corner ⇒ non-differentiable

Constant function derivative

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