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Maths formulas

Calculus

\[(uv)^{'} = u^{'}v + uv^{'}\]
\[\left(\frac{u}{v}\right)^{'} = \frac{u^{'}v - uv^{'}}{v^2}\]
\[\frac{dy}{dx} = \frac{dy} {du} \times \frac{du} {dx}\]

1. Polynomials

\[y = a\]
\[y' = 0\]
\[y = ax\]
\[y' = a\]
\[y = x^{n}\]
\[y' = nx^{n-1}\]
\[y = u^{n}\]
\[y' = nu^{n-1} \times u'\]

2. Trigonometric functions

\[y = \cos x \]
\[y'= -\sin x\]
\[y = \sin x \]
\[y'= \cos x\]
\[y = \cos (u) \]
\[y'= -\sin (u) \times u'\]
\[y = \sin (u) \]
\[y'= \cos (u) \times u'\]

3. Exponential functions

\[y = e^{x}\]
\[y' = e^{x}\]
\[y = a^{x}\]
\[y' = a^{x} \times \ln (a)\]
\[y = e^{u}\]
\[y' = e^{u} \times u'\]
\[y = a^{u}\]
\[y' = a^{u} \times \ln (a) \times u'\]

4. Logarithmic functions

\[y = \ln(x)\]
\[y' = \frac{1}{x}\]
\[y = \ln(u)\]
\[y' = \frac{u'}{u}\]

Note
\(\ln (x) \) is the same thing as \(\log_e (x) \)

1. Polynomials

\[y = a\]
\[\int a \enspace dx = ax + C\]
\[y = x^{n}\]
\[\int x^{n} \enspace dx = \frac{1}{n+1}x^{n+1} + C, n \ne -1\]

2. Trigonometric functions

\[y = \cos x \]
\[\int \cos x \enspace dx = \sin x + C\]
\[y = \sin x \]
\[\int \sin x \enspace dx = -\cos x + C\]
\[y = \cos (ax + b) \]
\[\int \cos (ax + b) \enspace dx = \frac{1}{a} \sin (ax + b) + C\]
\[y = \sin (ax + b) \]
\[\int \sin (ax + b) \enspace dx = -\frac{1}{a} \cos (ax + b) + C\]

3. Exponential functions

\[y = e^{x}\]
\[\int e^{x} \enspace dx = e^{x} + C\]
\[y = e^{ax + b}\]
\[\int e^{ax + b} \enspace dx = \frac{1}{a} e^{ax + b} + C \]

4. Logarithmic functions

\[y = \frac{1}{x}\]
\[\int \frac{1}{x} \enspace dx = \ln x + C \]
\[y = \frac{u'}{u}\]
\[\int \frac{u'}{u} \enspace dx = \ln (u) + C\]

Definite integrals

\[\begin{eqnarray*} \int_{a}^{b} f(x) \enspace dx &=& [F(x)]_{a}^{b} \\ &=& F(a) - F(b) \end{eqnarray*}\]

Note
\(\ln (x) \) is the same thing as \(\log_e (x) \)