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 | x^{2} | 2x |
Square Root | √x | (½)x^{-½} |
Exponential | e^{x} | e^{x} |
a^{x} | ln(a) a^{x} | |
Logarithms | ln(x) | 1/x |
log_{a}(x) | 1 / (x ln(a)) | |
Trigonometry (x is in radians) | sin(x) | cos(x) |
cos(x) | −sin(x) | |
tan(x) | sec^{2}(x) | |
Inverse Trigonometry | sin^{-1}(x) | 1/√(1−x^{2}) |
cos^{-1}(x) | −1/√(1−x^{2}) | |
tan^{-1}(x) | 1/(1+x^{2}) | |
Derivative Calculator
Rules | Function | Derivative |
---|---|---|
Multiplication by constant | cf | cf’ |
Power Rule | x^{n} | nx^{n−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 )/g^{2} |
Reciprocal Rule | 1/f | −f’/f^{2} |
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 |