What Is Derivative?
In mathematics, the “derivative” measures the sensitivity to change of output value with respect to a change in the input value but in calculus, derivatives are central tools.
Did You Know!
Many statisticians have defined derivatives simply by the following formula:
- d/dx∗f=f∗(x)=limh→0f(x+h)−f(x)/h
| Common Functions | Function | Derivative |
|---|---|---|
| Constant | c | 0 |
| Line | x | 1 |
| ax | a | |
| Square | x2 | 2x |
| Square Root | √x | (½)x-½ |
| Exponential | ex | ex |
| ax | ln(a) ax | |
| Logarithms | ln(x) | 1/x |
| loga(x) | 1 / (x ln(a)) | |
| Trigonometry (x is in radians) | sin(x) | cos(x) |
| cos(x) | −sin(x) | |
| tan(x) | sec2(x) | |
| Inverse Trigonometry | sin-1(x) | 1/√(1−x2) |
| cos-1(x) | −1/√(1−x2) | |
| tan-1(x) | 1/(1+x2) | |
Derivative Calculator
| Rules | Function | Derivative |
|---|---|---|
| Multiplication by constant | cf | cf’ |
| Power Rule | xn | nxn−1 |
| Sum Rule | f + g | f’ + g’ |
| Difference Rule | f – g | f’ − g’ |
| Product Rule | fg | f g’ + f’ g |
| Quotient Rule | f/g | (f’ g − g’ f )/g2 |
| Reciprocal Rule | 1/f | −f’/f2 |
| Chain Rule (as “Composition of Functions”) |
f º g | (f’ º g) × g’ |
| Chain Rule (using ’ ) |
f(g(x)) | f’(g(x))g’(x) |
| Chain Rule (using dydxdydx) |
dydx=dydududxdydx=dydududx | |