CBSE · Class 12 · Mathematics · Chapter 5

Continuity and Differentiability — Formula Sheet

Board Formulas

14 formulas

1.Definition of Derivative
$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
First-principles definition. Rarely asked directly, but conceptually foundational.
2.Continuity at a Point★
$lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = f (a)$
All three must be equal AND finite. If any one differs or is undefined, f is discontinuous at a.
3.Differentiability implies Continuity
$f differentiable at a \Rightarrow f continuous at a$
Converse is NOT true. |x| is continuous at 0 but not differentiable there.
4.Sum, Difference, Constant Multiple
$(u \pm v)^{'} = u^{'} \pm v^{'}, (c u)^{'} = c u^{'}$
Linearity of differentiation. Differentiate term-by-term.
5.Product Rule★
$\frac{d}{d x} (uv) = u^{'} v + u v^{'}$
First × derivative of second + second × derivative of first. NOT u'v'.
6.Quotient Rule★
$\frac{d}{d x} (\frac{u}{v}) = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$
Low d-high minus high d-low, over low squared. Order matters — sign errors are costly.
7.Chain Rule★
$\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$
Differentiate outer function keeping inner intact, then multiply by derivative of inner. Peel layers one at a time.
8.Standard Derivatives (Trig)★
$(sin x)^{'} = cos x, (cos x)^{'} = - sin x, (tan x)^{'} = sec^{2} x$
Cos gets a MINUS sign. cot, cosec also carry the minus.
9.Standard Derivatives (Inverse Trig)★
$(sin^{- 1} x)^{'} = \frac{1}{1 - x ^{2}}, (tan^{- 1} x)^{'} = \frac{1}{1 + x ^{2}}$
cos⁻¹ and cot⁻¹ carry an extra MINUS sign versus their sin⁻¹, tan⁻¹ counterparts.
10.Exponential and Logarithm
$(e^{x})^{'} = e^{x}, (a^{x})^{'} = a^{x} ln a, (ln x)^{'} = \frac{1}{x}$
e^x is its own derivative. General a^x picks up a factor of ln a.
11.Logarithmic Differentiation★
$y = [f (x)]^{g (x)} \Rightarrow ln y = g (x) ln f (x)$
Use when base AND exponent are variable, e.g. x^x, (sin x)^x. Then differentiate implicitly.
12.Parametric Differentiation★
$\frac{d y}{d x} = \frac{d y / d t}{d x / d t}$
When x = f(t), y = g(t), never write dy/dt · dx/dt — it is a QUOTIENT of the two.
13.Second Derivative
$\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{d y}{d x})$
Differentiate dy/dx once more with respect to x. Do not divide.
14.Implicit Differentiation
$\frac{d}{d x} (y^{n}) = n y^{n - 1} \frac{d y}{d x}$
For equations not solved for y (e.g. x² + y² = 25), differentiate both sides, treat y as function of x, then solve for dy/dx.