Derivative Rules
Basic Rules
| Rule | Formula | Example |
|---|---|---|
| Constant Rule | d/dx [c] = 0 | d/dx [5] = 0 |
| Power Rule | d/dx [xโฟ] = nยทxโฟโปยน | d/dx [xยณ] = 3xยฒ |
| Constant Multiple Rule | d/dx [cยทf] = cยทf' | d/dx [5xยฒ] = 10x |
| Sum/Difference Rule | d/dx [f ยฑ g] = f' ยฑ g' | d/dx [xยฒ + x] = 2x + 1 |
| Product Rule | d/dx [fยทg] = f'ยทg + fยทg' | d/dx [xยทsin(x)] = sin(x) + xยทcos(x) |
| Quotient Rule | d/dx [f/g] = (f'g - fg') / gยฒ | d/dx [x/eหฃ] = (eหฃ - xeหฃ) / eยฒหฃ |
| Chain Rule | d/dx [f(g(x))] = f'(g(x))ยทg'(x) | d/dx [sin(xยฒ)] = cos(xยฒ)ยท2x |
Common Function Derivatives
| f(x) | f'(x) |
|---|---|
| xโฟ | nยทxโฟโปยน |
| eหฃ | eหฃ |
| aหฃ | aหฃ ยท ln(a) |
| ln(x) | 1/x |
| log_a(x) | 1 / (x ยท ln(a)) |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | secยฒ(x) |
| cot(x) | -cscยฒ(x) |
| sec(x) | sec(x)ยทtan(x) |
| csc(x) | -csc(x)ยทcot(x) |
| arcsin(x) | 1 / โ(1 - xยฒ) |
| arccos(x) | -1 / โ(1 - xยฒ) |
| arctan(x) | 1 / (1 + xยฒ) |
| |x| | x / |x|, x โ 0 |