Derivatives
(i) Constant Function : derivates ZERO(0)
$\frac{dy}{dx}=\ 0$
$\frac{d}{dx}\ (2)=\ 0$
(ii) Polynomail Function
$\frac{d}{dx}(x^n) = nx^{n-1}$
(iii) Trigonometric Function
$\frac{d}{dx} \sin(x) = \ cos(x)$
$\frac{d}{dx} \cos(x) =-\sin(x)$
$\frac{d}{dx} \tan(x) = \ sec^2(x)$
$\frac{d}{dx} \cot(x) =-\ cosec^2(x)$
$\frac{d}{dx} \sec(x) = \ sec(x).tan(x)$
$\frac{d}{dx} \ cosec(x) =-\ cosec(x).cot(x)$
(iv)Log Function
$\frac{d}{dx}\ log(x)=\frac{1}{x}$
(v)Exponential Function
$\frac{d}{dx}\ (e^x)=e^x$
Product Rule $\frac{d}{dx} (uv)=u\frac{dv}{dx}+v.\frac{du}{dx}$
Quotient Rule $ \frac{d}{dx} \left( \frac{u}{v} \right)= \frac{v . \frac{du}{dx} - u . \frac{dv}{dx}}{v^2}$
Inverse Function $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}(\cos^{-1} x) =-\frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}(\tan^{-1} x) =\frac{1}{1+x^2}$ $\frac{d}{dx}(\cot^{-1} x) =-\frac{1}{1+x^2}$ $\frac{d}{dx}(\sec^{-1} x) =\frac{1}{x\sqrt{x^2-1}}$ $\frac{d}{dx}(\ cosec^{-1} x) =-\frac{1}{x\sqrt{x^2-1}}$ Hyperbolic Function $\sinh x = \frac{e^x - e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2}$ $\frac{d}{dx}(\sinh x) = \cosh x$ $\frac{d}{dx}(\cosh x) = \sinh x$ (Note: No negative sign here!) $\frac{d}{dx}(\tanh x) = \text{sech}^2 x$ $\frac{d}{dx}(\coth x) = -\text{cosech}^2 x$ $\frac{d}{dx}(\text{sech } x) = -\text{sech } x \tanh x$ $\frac{d}{dx}(\text{cosech } x) = -\text{cosech } x \coth x$
Inverse Hyperbolic Function $\frac{d}{dx}(\sinh^{-1} x) = \frac{1}{\sqrt{x^2+1}}$ $\frac{d}{dx}(\cosh^{-1} x) = \frac{1}{\sqrt{x^2-1}}, \quad x > 1$ $\frac{d}{dx}(\tanh^{-1} x) = \frac{1}{1-x^2}, \quad x<1$ $\frac{d}{dx}(\coth^{-1} x) = \frac{1}{1-x^2}, \quad |x| > 1$ $\frac{d}{dx}(\text{sech}^{-1} x) = -\frac{1}{x\sqrt{1-x^2}}, \quad 0 < x < 1$ $\frac{d}{dx}(\text{csch}^{-1} x) = -\frac{1}{|x|\sqrt{1+x^2}}, \quad x \neq 0$
Class 11 - Antiderivatives Formula ||Algebraic Function|| 1) $\int x^n \cdot dx = \frac{x^{n+1}}{n+1} + C, \quad (n \neq -1)$ 2) $\int x^{-1} \cdot dx = \int \frac{1}{x} \cdot dx = \log x + C$ 3) $\int 1 \cdot dx = x + C$ 4) $\int (ax+b)^n \cdot dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C, \quad (n \neq -1)$ 5) $\int (ax+b)^{-1} \cdot dx = \int \frac{1}{ax+b} \cdot dx = \frac{\log(ax+b)}{a} + C$ 6) $\int e^{ax+b} \cdot dx = \frac{e^{ax+b}}{a} + C$
Trigonometric Function Antiderivative 1) $\int \sin x \, dx = -\cos x + C$ 2) $\int \cos x \, dx = \sin x + C$ 3) $\int \sec^2 x \, dx = \tan x + C$ 4) $\int \ cosec^2 x \, dx = -\cot x + C$ 5) $\int \sec x \tan x \, dx = \sec x + C$ 6) $\int \ cosec x \cot x \, dx = -\ cosec x + C$ 7) $\int \tan x \, dx = \log |\sec x| + C$ 8) $\int \cot x \, dx = \log |\sin x| + C$ 9) $\int \sec x \, dx = \log |\sec x + \tan x| + C$ 10) $\int \ cosec x \, dx = \log |\ cosec x - \cot x| + C$
CLASS - 12 (Standard Integrals) $\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$ $\int \frac{1}{x^2 - a^2} dx = \frac{1}{2a} \log ( \frac{x-a}{x+a} ) + C$ $\int \frac{1}{a^2 - x^2} dx = \frac{1}{2a} \log ( \frac{a+x}{a-x} )+ C$ $\int \frac{1}{\sqrt{x^2 + a^2}} dx = \log (x + \sqrt{x^2 + a^2}) + C$ $\int \frac{1}{\sqrt{x^2 - a^2}} dx = \log (x + \sqrt{x^2 - a^2}) + C$ $\int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C$
Product Rule $\frac{d}{dx} (uv)=u\frac{dv}{dx}+v.\frac{du}{dx}$
Quotient Rule $ \frac{d}{dx} \left( \frac{u}{v} \right)= \frac{v . \frac{du}{dx} - u . \frac{dv}{dx}}{v^2}$
Inverse Function $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}(\cos^{-1} x) =-\frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}(\tan^{-1} x) =\frac{1}{1+x^2}$ $\frac{d}{dx}(\cot^{-1} x) =-\frac{1}{1+x^2}$ $\frac{d}{dx}(\sec^{-1} x) =\frac{1}{x\sqrt{x^2-1}}$ $\frac{d}{dx}(\ cosec^{-1} x) =-\frac{1}{x\sqrt{x^2-1}}$ Hyperbolic Function $\sinh x = \frac{e^x - e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2}$ $\frac{d}{dx}(\sinh x) = \cosh x$ $\frac{d}{dx}(\cosh x) = \sinh x$ (Note: No negative sign here!) $\frac{d}{dx}(\tanh x) = \text{sech}^2 x$ $\frac{d}{dx}(\coth x) = -\text{cosech}^2 x$ $\frac{d}{dx}(\text{sech } x) = -\text{sech } x \tanh x$ $\frac{d}{dx}(\text{cosech } x) = -\text{cosech } x \coth x$
Inverse Hyperbolic Function $\frac{d}{dx}(\sinh^{-1} x) = \frac{1}{\sqrt{x^2+1}}$ $\frac{d}{dx}(\cosh^{-1} x) = \frac{1}{\sqrt{x^2-1}}, \quad x > 1$ $\frac{d}{dx}(\tanh^{-1} x) = \frac{1}{1-x^2}, \quad x<1$ $\frac{d}{dx}(\coth^{-1} x) = \frac{1}{1-x^2}, \quad |x| > 1$ $\frac{d}{dx}(\text{sech}^{-1} x) = -\frac{1}{x\sqrt{1-x^2}}, \quad 0 < x < 1$ $\frac{d}{dx}(\text{csch}^{-1} x) = -\frac{1}{|x|\sqrt{1+x^2}}, \quad x \neq 0$
Class 11 - Antiderivatives Formula ||Algebraic Function|| 1) $\int x^n \cdot dx = \frac{x^{n+1}}{n+1} + C, \quad (n \neq -1)$ 2) $\int x^{-1} \cdot dx = \int \frac{1}{x} \cdot dx = \log x + C$ 3) $\int 1 \cdot dx = x + C$ 4) $\int (ax+b)^n \cdot dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C, \quad (n \neq -1)$ 5) $\int (ax+b)^{-1} \cdot dx = \int \frac{1}{ax+b} \cdot dx = \frac{\log(ax+b)}{a} + C$ 6) $\int e^{ax+b} \cdot dx = \frac{e^{ax+b}}{a} + C$
Trigonometric Function Antiderivative 1) $\int \sin x \, dx = -\cos x + C$ 2) $\int \cos x \, dx = \sin x + C$ 3) $\int \sec^2 x \, dx = \tan x + C$ 4) $\int \ cosec^2 x \, dx = -\cot x + C$ 5) $\int \sec x \tan x \, dx = \sec x + C$ 6) $\int \ cosec x \cot x \, dx = -\ cosec x + C$ 7) $\int \tan x \, dx = \log |\sec x| + C$ 8) $\int \cot x \, dx = \log |\sin x| + C$ 9) $\int \sec x \, dx = \log |\sec x + \tan x| + C$ 10) $\int \ cosec x \, dx = \log |\ cosec x - \cot x| + C$
CLASS - 12 (Standard Integrals) $\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$ $\int \frac{1}{x^2 - a^2} dx = \frac{1}{2a} \log ( \frac{x-a}{x+a} ) + C$ $\int \frac{1}{a^2 - x^2} dx = \frac{1}{2a} \log ( \frac{a+x}{a-x} )+ C$ $\int \frac{1}{\sqrt{x^2 + a^2}} dx = \log (x + \sqrt{x^2 + a^2}) + C$ $\int \frac{1}{\sqrt{x^2 - a^2}} dx = \log (x + \sqrt{x^2 - a^2}) + C$ $\int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C$
