Differentiability formulas

Every formula JEE Advanced expects you to know for Differentiability, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 26
Subject: Mathematics
Exam: JEE Advanced
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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
Right-hand derivative
$f^{'} (a^{+}) = h \to 0 lim \frac{f ( a + h ) - f ( a )}{h}$
Left-hand derivative
$f^{'} (a^{-}) = h \to 0 lim \frac{f ( a - h ) - f ( a )}{- h}$
Differentiability condition
$f^{'} (a^{+}) = f^{'} (a^{-}) (both finite)$
Differentiable ⇒ continuous
$f differentiable at a \Rightarrow f continuous at a$
converse false
Discontinuous ⇒ non-differentiable
$f discontinuous at a \Rightarrow f not differentiable at a$
Differentiable on (a,b)
$differentiable at each point of (a, b)$
Differentiable on [a,b]
$diff. on (a, b), right-deriv at a, left-deriv at b$
Sum of differentiable
$f \pm g differentiable$
if f,g differentiable
Product of differentiable
$f \cdot g differentiable$
Diff ± non-diff
$differentiable \pm non-differentiable = non-differentiable$
Polynomials differentiable
$polynomials, e^{x}, sin, cos, lo g differentiable on their domains$
|x-a| non-differentiable
$∣ x - a ∣ is not differentiable at x = a$
sharp corner
(x-a)^n|x-a|
$(x - a)^{n} ∣ x - a ∣ differentiable at a iff n \geq 1$
sgn(x-a) non-differentiable
$sgn (x - a) not differentiable at x = a$
jump
xⁿ sin(1/x) differentiability
$x^{n} sin \frac{1}{x} differentiable at 0 iff n > 1$
{x},[x] at integers
${x}, [x] not differentiable at integers$
Leibnitz differentiation
$\frac{d}{d x} \int_{α (x)}^{β (x)} f (t) d t = f (β) β^{'} - f (α) α^{'}$
Vertical tangent
$f continuous, ∣ f^{'} (x) ∣ \to \infty at a \Rightarrow vertical tangent, non-diff.$
Oscillation point
$high-frequency oscillation \Rightarrow non-differentiable$
FE f(x+y)=f(x)f(y)
$f (x + y) = f (x) f (y) \Rightarrow f^{'} (x) = f^{'} (0) f (x)$
FE f(x+y)=f(x)+f(y)
$f (x + y) = f (x) + f (y) \Rightarrow f (x) = k x$
FE f(xy)=f(x)+f(y)
$f (x y) = f (x) + f (y) \Rightarrow f (x) = k lo g x$
FE f(x)f(1/x)=f(x)+f(1/x)
$f (x) f (\frac{1}{x}) = f (x) + f (\frac{1}{x}) \Rightarrow f (x) = 1 \pm x^{n}$
Corner ⇒ non-differentiable
$sharp corner (LHD \neq = RHD) \Rightarrow non-differentiable$
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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