Online Calculators & Tools
Derivative rules and laws. Derivatives of functions table.
The derivative of a function is the ratio of the difference of function value f(x) at points x+Δx and x with Δx, when Δx is infinitesimally small. The derivative is the function slope or slope of the tangent line at point x.
The second derivative is given by:
Or simply derive the first derivative:
The nth derivative is calculated by deriving f(x) n times.
The nth derivative is equal to the derivative of the (n-1) derivative:
f^{ (n)}(x) = [f^{(n-1)}(x)]'
The derivative of a function is the slop of the tangential line.
Derivative sum rule | ( a f (x) + bg(x) ) ' = a f ' (x) + bg' (x) |
Derivative product rule | ( f (x) ∙ g(x) ) ' = f ' (x) g(x) + f (x) g' (x) |
Derivative quotient rule | |
Derivative chain rule | f ( g(x) ) ' = f ' (g(x) ) ∙ g' (x) |
When a and b are constants.
( a f (x) + bg(x) ) ' = a f ' (x) + bg' (x)
For small Δx, we can get an approximation to f(x_{0}+Δx), when we know f(x_{0}) and f ' (x_{0}):
f (x_{0}+Δx) ≈ f (x_{0}) + f '(x_{0})⋅Δx
Function name | Function | Derivative |
---|---|---|
f (x) | f '(x) | |
Constant | const | 0 |
Linear | x | 1 |
Power | x^{ a} | a x^{ a-}^{1} |
Exponential | e^{ x} | e^{ x} |
Exponential | a^{ x} | a^{ x }ln a |
Natural logarithm | ln(x) | |
Logarithm | log_{b}(x) | |
Sine | sin x | cos x |
Cosine | cos x | -sin x |
Tangent | tan x | |
Arcsine | arcsin x | |
Arccosine | arccos x | |
Arctangent | arctan x | |
Hyperbolic sine | sinh x | cosh x |
Hyperbolic cosine | cosh x | sinh x |
Hyperbolic tangent | tanh x | |
Inverse hyperbolic sine | sinh^{-1} x | |
Inverse hyperbolic cosine | cosh^{-1} x | |
Inverse hyperbolic tangent | tanh^{-1} x |
When the first derivative of a function is zero at point x_{0}.
f '(x_{0}) = 0
Then the second derivative at point x_{0} , f''(x_{0}), can indicate the type of that point:
f ''(x_{0}) > 0 | local minimum |
f ''(x_{0}) < 0 | local maximum |
f ''(x_{0}) = 0 | undetermined |
Derivatives are a cornerstone in calculus, providing a powerful tool for understanding how a function changes. Calculating derivatives involves finding the rate at which a function changes with respect to its independent variable. Whether you're a student diving into calculus or a professional using mathematical tools, understanding derivatives is crucial.
This fundamental concept finds applications in physics, economics, engineering, and more. Whether analyzing the motion of objects, modeling financial trends, or optimizing processes, derivatives play a key role.
Calculating derivatives involves various techniques, such as the power rule, product rule, quotient rule, and chain rule. Our guide breaks down these methods, offering step-by-step explanations and examples to facilitate a clear understanding. Dive into the world of derivatives with our comprehensive guide and unlock their potential in mathematical problem-solving.