Notation and Rules for Differentiating Functions

As already known the commonly used symbols for functions are \(f(x),\) and \(g(x).\) These two functions represnet functions of \(x\). The value of the function \(f(x)\) at \(x=b\) is denoted as \(f(b)\). The derivative of the function \(y=f(x)\) or the derivative of \(y\) with respect to \(x\) is usually denoted by one of the following $$ \frac{\mathrm{d}y}{\mathrm{d}x}, f'(x), y'.$$ In previous notation only the first derivative of \(y\) with respect to \(x\) is shown. The second and third derivative of \(y\) with respect to \(x\) can be written in the following form $$ \frac{\mathrm{d}^2y}{\mathrm{d}x^2} = \frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right) = \frac{\mathrm{d}}{\mathrm{d}x}f'(x) = f''(x),$$ $$ \frac{\mathrm{d}^3y}{\mathrm{d}x^3} = \frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\right) = \frac{\mathrm{d}}{\mathrm{d}x}f''(x) = f'''(x).$$ The values of the second and thrid derivative of function \(f(x)\) at \(x = b\) are denoted as \(f''(b)\), and \(f''(b).\)

The basic rules of derivatives - if \(c\) is constant, and \(y = \phi(x)\), \(z = \psi(x)\) are functions then derivatives of function

1. \((c)' = 0,\)
2. \((x)' = 1,\)
3. \((y\pm z)' = y' \pm z',\)
4. \((cy)' = cy'\)
5. \((yz)' = y'z + yz'\)
6. \(\left(\frac{y}{z}\right)' = \frac{y'z-yz'}{z^2} \quad v\neq 0\)
7. \(\left(\frac{c}{z}\right)' = -\frac{cv'}{v^2}\quad v\neq0.\)

Table of basic function derivates

1. \((x^n)' = nx^{n-1},\)
2. \(\left(\sqrt{x}\right)' = \frac{1}{2\sqrt{x}} \quad x > 0,\)
3. \((\sin x)' = \cos x,\)
4. \((\cos x)' = -\sin x,\)
5. \((\tan x)' = \frac{1}{\cos^2 x},\)
6. \((\cot x)' = -\frac{1}{\sin^2 x},\)
7. \((\arcsin x)' = \frac{1}{\sqrt{1-x^2}}, |x| < 1 \)
8. \((\arccos x)' = -\frac{1}{\sqrt{1-x^2}}, |x| < 1 \)
9. \((\arctan x)' = \frac{1}{1+x^2}\)
10. \((\mathrm{arccot} x)' = -\frac{1}{1+x^2}\)
11. \((a^x)' = a^x \ln a \quad a > 0\)
12. \((e^x)' = e^x,\)
13. \((\ln x)' = \frac{1}{x} x > 0\)
14. \((\log_a x )' = \frac{1}{x \ln a} = \frac{\log_a e}{x}, \)
15. \((\sinh x)' = \cosh x,\)
16. \((\cosh x)' = \sinh x, \)
17. \((\tanh x)' = \frac{1}{\cosh^2 x}\)
18. \((\coth x)' = -\frac{1}{\sinh^2 x}\)
19. \((\mathrm{arcsinh} x)' = \frac{1}{\sqrt{1+x^2}}\)
20. \((\mathrm{arccosh} x)' = \frac{1}{\sqrt{x^2-1}} \quad x > 1\)
21. \((\mathrm{arctanh} x)' = \frac{1}{1 -x^2 } |x| < 1 \quad x > 0, a > 0 \)
22. \((\mathrm{arctanh} x)' = -\frac{1}{x^2 - 1} |x| > 1\)