Derivatives
| Function | Derivative |
|---|---|
| \(k\) (constant) | \(\frac{d}{dx}(k) = 0\) |
| \(x\) | \(\frac{d}{dx}(x) = 1\) |
| \(x^2\) | \(\frac{d}{dx}(x^2) = 2x\) |
| \(x^3\) | \(\frac{d}{dx}(x^3) = 3x^2\) |
| \(x^n\) (for any constant \(n\)) | \(\frac{d}{dx}(x^n) = nx^{n-1}\) |
| \(e^x\) | \(\frac{d}{dx}(e^x) = e^x\) |
| \(e^{kx}\) | \(\frac{d}{dx}(e^{kx}) = ke^{kx}\) |
| \(\ln x = \log_e x\) | \(\frac{d}{dx}(\ln x) = \frac{1}{x}\) |
| \(\sin x\) | \(\frac{d}{dx}(\sin x) = \cos x\) |
| \(\cos x\) | \(\frac{d}{dx}(\cos x) = -\sin x\) |
| \(\tan x\) | \(\frac{d}{dx}(\tan x) = \sec^2 x\) |
| \(\csc x\) | \(\frac{d}{dx}(\csc x) = -\csc x \cot x\) |
| \(\cot x\) | \(\frac{d}{dx}(\cot x) = -\csc^2 x\) |
| \(\sec x\) | \(\frac{d}{dx}(\sec x) = \sec x \tan x\) |
| \(\sin^{-1} x\) | \(\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}}\) |
| \(\cos^{-1} x\) | \(\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}}\) |
| \(\tan^{-1} x\) | \(\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2}\) |
| \(\sinh x\) | \(\frac{d}{dx}(\sinh x) = \cosh x\) |
| \(\cosh x\) | \(\frac{d}{dx}(\cosh x) = \sinh x\) |
| \(\tanh x\) | \(\frac{d}{dx}(\tanh x) = \mathrm{sech}^2 x\) |
Important Properties of Derivatives
- \[\int f(x) \, dx = F(x) + C,\]is the constant of integration. where \(f(x)\)\(F(x)\)First Fundamental Rule of Calculus:
- \[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.\]\(u = f(x)\)\(y = g(u)\)Chain Rule:
Integrals
| Function | Integral |
|---|---|
| \(\int u^n \, du\) | \(\frac{u^{n+1}}{n+1} + C, \quad n \neq -1\) |
| \(\int \frac{du}{u}\) | \(\ln |u| + C\) |
| \(\int e^u \, du\) | \(e^u + C\) |
| \(\int au \, du\) | \(\frac{a}{\ln a} \ln |u| + C\) |
| \(\int \sin u \, du\) | \(-\cos u + C\) |
| \(\int \cos u \, du\) | \(\sin u + C\) |
| \(\int \sec^2 u \, du\) | \(\tan u + C\) |
| \(\int \csc^2 u \, du\) | \(-\cot u + C\) |
| \(\int \sec u \tan u \, du\) | \(\sec u + C\) |
| \(\int \csc u \cot u \, du\) | \(-\csc u + C\) |
| \(\int \tan u \, du\) | \(\ln |\sec u| + C\) |
| \(\int \cot u \, du\) | \(\ln |\sin u| + C\) |
| \(\int \sec u \, du\) | \(\ln |\sec u + \tan u| + C\) |
| \(\int \csc u \, du\) | \(\ln |\csc u - \cot u| + C\) |
| \(\int \frac{du}{\sqrt{a^2 - u^2}}\) | \(\arcsin\left(\frac{u}{a}\right) + C\) |
| \(\int \frac{du}{a^2 + u^2}\) | \(\frac{1}{a} \arctan\left(\frac{u}{a}\right) + C\) |
| \(\int \frac{du}{|u|\sqrt{u^2 - a^2}}\) | \(\frac{1}{a} \text{arcsec}\left(\frac{|u|}{a}\right) + C\) |
| \(\int \ln u \, du\) | \(u\ln u - u + C\) |
| \(\int \sinh u \, du\) | \(\cosh u + C\) |
| \(\int \cosh u \, du\) | \(\sinh u + C\) |
| \(\int \tanh u \, du\) | \(\ln(\cosh u) + C\) |
| \(\int \arcsin u \, du\) | \(u\arcsin u + \sqrt{1-u^2} + C\) |
| \(\int \arccos u \, du\) | \(u\arccos u - \sqrt{1-u^2} + C\) |
| \(\int \arctan u \, du\) | \(u\arctan u - \frac{1}{2}\ln(1+u^2) + C\) |
Properties of Integrals
- \[\int (af(u) + bg(u)) \, du = a\int f(u) \, du + b\int g(u) \, du\]Linearity Property:
- \[\int u \, dv = uv - \int v \, du\]Integration by Parts:
- \[\int k \cdot f(u) \, du = k\int f(u) \, du\]Constant Multiple Rule:
- \[\int (f(u) \pm g(u)) \, du = \int f(u) \, du \pm \int g(u) \, du\]Sum and Difference Rule:
- \[\int f(u) \, du = \int f(g(x)) \cdot g'(x) \, dx\]Change of Variable Rule:
- \[\int_a^a f(x) \, dx = 0\]Zero-Integral Rule:
- \[\int_a^b c \, dx = c(b - a)\]Integral of a Constant:
- \[\int_{-a}^a f(x) \, dx = 2\int_0^a f(x) \, dx\]Symmetry Rule:
- \[\left|\int_a^b f(x) \, dx\right| \leq \int_a^b |f(x)| \, dx\]Absolute Value Rule: