12Class 11 Maths

Limits and Derivatives formulas

Standard limits and derivative formulas. You'll use these all the way through Class 12 calculus.

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All Maths formulas

Standard limits

Polynomial form: $\displaystyle \lim_{x \to a}\frac{x^n - a^n}{x - a} = na^{n - 1}$
Sine: $\displaystyle \lim_{x \to 0}\frac{\sin x}{x} = 1$
x in radians.
Cosine: $\displaystyle \lim_{x \to 0}\frac{1 - \cos x}{x} = 0$
Tangent: $\displaystyle \lim_{x \to 0}\frac{\tan x}{x} = 1$

Derivatives

First principles: $\displaystyle f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$
Power: $\displaystyle \frac{d}{dx}x^n = nx^{n - 1}$
Trigonometric: $\displaystyle \frac{d}{dx}\sin x = \cos x, \quad \frac{d}{dx}\cos x = -\sin x, \quad \frac{d}{dx}\tan x = \sec^2x$
Product rule: $\displaystyle (uv)' = u'v + uv'$
Quotient rule: $\displaystyle \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

Where students slip up

Substituting straight away into a limit that gives 0/0. Simplify or factor first.
Using sin x / x → 1 with x in degrees. It only holds in radians.
Getting the order wrong in the quotient rule, since the minus makes it matter.

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