05Class 12 Maths

Continuity and Differentiability formulas

Continuity conditions and all the differentiation rules for Class 12. This is the calculus you'll use for the rest of the year.

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Continuity

Condition at a point: $\displaystyle \lim_{x \to c^-}f(x) = \lim_{x \to c^+}f(x) = f(c)$

Chain rule: $\displaystyle \frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$
Parametric form: $\displaystyle \frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
Logarithmic differentiation: $\displaystyle y = u^v \;\Rightarrow\; \frac{dy}{dx} = u^v\left[\frac{v}{u}\frac{du}{dx} + \ln u\,\frac{dv}{dx}\right]$

Exponential and log: $\displaystyle \frac{d}{dx}e^x = e^x, \quad \frac{d}{dx}\ln x = \frac1x, \quad \frac{d}{dx}a^x = a^x\ln a$
Other trigonometric: $\displaystyle \frac{d}{dx}\sec x = \sec x\tan x, \quad \frac{d}{dx}\csc x = -\csc x\cot x, \quad \frac{d}{dx}\cot x = -\csc^2x$
Inverse sine and cosine: $\displaystyle \frac{d}{dx}\sin^{-1}x = \frac{1}{\sqrt{1 - x^2}}, \quad \frac{d}{dx}\cos^{-1}x = -\frac{1}{\sqrt{1 - x^2}}$
Inverse tangent: $\displaystyle \frac{d}{dx}\tan^{-1}x = \frac{1}{1 + x^2}$

Assuming every continuous function is differentiable. |x| is continuous at 0 but isn't differentiable there.
Stopping the chain rule early with functions like sin(x²).
Using the power rule on xˣ. Take logs first.

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