$$ \begin{aligned} r^2&=x^2+y^2 \\ \theta&=\tan^{-1}(\frac{y}{x}) \\ x&=r*\cos{\theta} \\ y&=r*\sin{\theta} \end{aligned} $$
function are defined either as $f(\theta)=r \text{ or } f(r)=\theta$
| Definition: | Name: |
|---|---|
| $\frac{e^{iz}-e^{-iz}}{2i}=\text{Im}(e^{ix})$ | $\sin{z}$ |
| $\frac{e^{iz}+e^{-iz}}{2}=\text{Re}(e^{ix})$ | $\cos{z}$ |
| $\frac{\sin{x}}{\cos{x}}$ | $\tan{x}$ |
| $\frac{1}{\sin{x}}$ | $\csc{x}$ |
| $\frac{1}{\cos{x}}$ | $\sec{x}$ |
| $\frac{1}{\tan{x}}$ | $\cot{x}$ |
| $\csc({x})-1$ | $\text{excsc}\,x$ |
| $\sec({x})-1$ | $\text{exsec}\,x$ |
| $1-\cos{x} \\ \text{or } 2\sin^2(\frac{x}{2})$ | $\text{versin}\,x$ |
| $1-\sin{x} \\ \text{or versin} (\frac{\pi}{2}-x)$ | $\text{cvs}\,{x}$ |
| $\frac{e^x-e^{-x}}{2}$ | $\sinh{x}$ |
| $\frac{e^x+e^{-x}}{2}$ | $\cosh{x}$ |
| $\frac{\sinh{x}}{\cosh{x}}$ | $\tanh{x}$ |
| $\frac{1}{\sinh{x}}$ | $\text{csch}\,{x}$ |
| $\frac{1}{\cosh{x}}$ | $\text{sech}\,x$ |
| $\frac{1}{\tanh{x}}$ | $\text{coth}\,x$ |
| $f(x)$ | $\dot{f}(x)$ |
|---|---|
| $x^n$ | $nx^{n-1}$ |
| $\sin{x}$ | $\cos{x}$ |
| $\sin^{-1}{x}$ | $\frac{1}{\sqrt{1-x^2}}$ |
| $\cos{x}$ | $-\sin(x)$ |
| $\cos^{-1}{x}$ | $\frac{-1}{\sqrt{1-x^2}}$ |
| $\tan{x}$ | $\sec^2{x}$ |
| $\tan^{-1}{x}$ | $\frac{1}{1+x^2}$ |
| $e^x$ | $e^x$ |
| $a^x$ | $\ln(a)a^x$ |
| $\ln{x}$ | $\frac{1}{x}$ |
| $\ln{g(x)}$ | $\frac{\dot{g}(x)}{g(x)}$ |
| $\log_a{x}$ | $\frac{1}{x\ln{a}}$ |
| $\sinh{x}$ | $\cosh{x}$ |
| $\sinh^{-1}{x}$ | $\frac{1}{\sqrt{x^2+1}}$ |
| $\cosh{x}$ | $\sinh{x}$ |
| $\cosh^{-1}{x}$ | $\frac{1}{\sqrt{x^2-1}}$ |
| $\tanh{x}$ | $\text{sech}^2{x}$ |
| $\tanh^{-1}{x}$ | $\frac{1}{1-x^2}$ |
| $c*f(x)$ | $x*\dot{f}(x)$ |
| $f(x)\pm g(x)$ | $\dot{f}(x) \pm \dot{g}(x)$ |
| $f(x)*g(x)$ | $\dot{f}(x)g(x)+f(x)\dot{g}(x)$ |
| $\frac{f(x)}{g(x)}$ | $\frac{\dot{f}(x)g(x)-f(x)\dot{g}(x)}{g(x)^2}$ |
| $f(g(x))$ or $\frac{\text{d}f}{\text{d}x}$ | $\dot{f}(g(x))\dot{g}(x)$ or $\frac{\text{d}f}{\text{d}g}\frac{\text{d}g}{\text{d}x}$ |
example:
$$ \begin{aligned} \frac{\text{d}}{\text{d}x}&[x^2+4x+3xy+y^3-6=0]_y \\ &=2x+4+3(y+x\dot{y})+3y^2\dot{y}=0\\ \frac{\text{d}y}{\text{d}x}&=-\frac{3y+2x+4}{2y^2+3x} \end{aligned} $$
$$ \frac{\text{d}y}{\text{d}x}=\frac{\text{d}y}{\text{d}t}\frac{\text{d}t}{\text{d}x}=\frac{\text{d}y}{\text{d}t}\left(\frac{\text{d}x}{\text{d}t}\right)^{-1} $$
example:
$$ \begin{aligned} &x= 4\cos{t},\;\; y=2\sin{t}\\ \frac{\text{d}y}{\text{d}x}&=\frac{\text{d}y}{\text{d}t}\left(\frac{\text{d}x}{\text{d}t}\right)^{-1} \\ &=2\cos{t}*\frac{1}{-4\sin{t}} \\ &=-\frac{1}{2}\cot{x} \end{aligned} $$
$$ \frac{\text{d}\ln[f(x)]}{\text{d} x}=\frac{\dot{f}(x)}{f(x)} $$
example:
$$ \begin{aligned} \text{funct}&\text{ion: }y=a^x \\ \frac{\text{d} (\ln y)}{\text{d} x} &= \frac{\text{d} (x\ln a)}{\text{d} x}\\ &= \ln a = \frac{1}{y}\frac{\text{d} y}{\text{d} x}\\ \text{so: } \frac{\text{d} y}{\text{d} x}&= y\ln a\\ &= a^{x}\ln a \end{aligned} $$