Use ^ for powers, * for multiplication, sin(), cos(), tan(), ln(), exp(), sqrt()
Power Rule
d/dx(x^n) = n·x^(n-1)
Product Rule
d/dx(u·v) = u'v + uv'
Quotient Rule
d/dx(u/v) = (u'v - uv')/v²
Chain Rule
d/dx(f(g(x))) = f'(g(x))·g'(x)
Powers: x^2, x^3, (x+1)^2
Multiplication: 3*x, 2*x^2
Functions: sin(x), cos(x), tan(x), ln(x), exp(x), sqrt(x)
Examples:
- x^3 + 2*x^2 - 5*x + 1
- sin(x) + cos(x)
- x^2 * exp(x)
- ln(x^2 + 1)
A derivative represents the instantaneous rate of change of a function with respect to one of its variables. In geometric terms, the derivative at a point gives the slope of the tangent line to the function's graph at that point. The derivative is a fundamental concept in calculus and is used extensively in physics, engineering, economics, and many other fields to model rates of change and optimize functions.
The notation f'(x) or df/dx represents the first derivative, f''(x) or d²f/dx² represents the second derivative (rate of change of the rate of change), and so on. Higher-order derivatives are useful for understanding acceleration, concavity, and other properties of functions.
Physics
Velocity is the derivative of position, and acceleration is the derivative of velocity. Derivatives help describe motion, forces, and energy transformations.
Economics
Marginal cost, marginal revenue, and elasticity are all calculated using derivatives to analyze how quantities change with respect to price or output.
Optimization
Finding maximum and minimum values of functions by setting the derivative equal to zero is essential in engineering design and resource allocation.
Machine Learning
Gradient descent algorithms use derivatives to minimize loss functions and train neural networks by iteratively adjusting parameters.
Derivative calculations follow standard calculus rules. Results depend on correct function input and selected derivative order. This calculator handles common functions and may not support all mathematical expressions. For complex or specialized functions, please verify results with additional resources.