Calculus • Derivatives & Integrals
Derivative & Integral Calculator
Numerical calculus for any function. Enter expressions like sin(x), x^2, exp(x), or log(x).
f(x) = sin(x)
Derivative at x = 1
0.5403
f'(1) ≈ 0.5403 using central difference with h = 0.001
Definite Integral
2.0000
∫₀^π sin(x) dx ≈ 2.0000 using Simpson's rule (500 steps)
📖 Supported Functions
sin(x)cos(x)tan(x)
exp(x)log(x)sqrt(x)
x^nabs(x)asin(x)
acos(x)atan(x)sinh(x)
Use ^ for exponent, * for multiplication
What This Derivative & Integral Calculator Does
Enter a polynomial expression and this tool computes both its derivative (the rate of change) and its indefinite integral (the antiderivative), showing the term-by-term steps using the power rule.
How It Works
Power Rule for Derivatives: the derivative of axn is n·axn-1. Apply this term by term to differentiate a polynomial.
Power Rule for Integrals: the integral of axn is (a/(n+1))xn+1 + C, where C is the constant of integration (since any constant's derivative is zero, it can't be recovered from the derivative alone).
What Derivatives and Integrals Represent
A derivative measures instantaneous rate of change — for a position function, the derivative is velocity. An integral measures accumulated area under a curve — for a velocity function, the integral is total distance traveled. They are inverse operations of each other, which is the core idea behind the Fundamental Theorem of Calculus.
Frequently Asked Questions
What is the power rule?
For a term axⁿ, the derivative is n·axⁿ⁻¹ — multiply by the exponent, then reduce the exponent by one. It's the most common rule used to differentiate polynomials.
Why does an integral include a '+ C'?
Because the derivative of any constant is zero, an infinite family of functions differing only by a constant all share the same derivative. The '+ C' acknowledges that, without more information (like a known point on the curve), the constant can't be determined.
How are derivatives and integrals related?
They are inverse operations: differentiating a function's integral returns the original function. This relationship is formalized by the Fundamental Theorem of Calculus, which connects rates of change to accumulated totals.