Derivative Calculator
Calculate derivatives and rates of change using the power rule.
Function Settings
Supported: x^n, sin(x), cos(x), tan(x), e^x, ln(x), sqrt(x)
Derivative Rules
- • Power Rule: d/dx[x^n] = n·x^(n-1)
- • Constant Rule: d/dx[c] = 0
- • Sum Rule: d/dx[f+g] = f' + g'
- • Product Rule: d/dx[f·g] = f'·g + f·g'
- • Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
Derivative Expression
d/dx[x^2 + 3*x + 2]
Numerical Value
At x = 1
Solution Steps
How it works
A derivative measures the instantaneous rate of change of a function — the slope of its graph at a point. The power rule handles polynomials: bring the exponent down as a multiplier and reduce it by one. It's the foundation of calculus, used for velocities, optimization, and more.
Power rule
d/dx [ xⁿ ] = n · x^(n−1) constants differentiate to 0
- n
- the exponent
- x
- the variable
Worked example
- Differentiate f(x) = 3x² + 5x
- d/dx[3x²] = 6x
- d/dx[5x] = 5
f'(x) = 6x + 5.
Good to know
- The derivative gives the slope of the tangent line at each point.
- Where the derivative is zero, the function has a flat point — a max, min, or inflection.
- Product, quotient, and chain rules extend differentiation to combined functions.
Verwandte Rechner
Häufig gestellte Fragen
What is a derivative?
The instantaneous rate of change of a function — geometrically, the slope of its graph at a point. If position is f(t), the derivative f′(t) is velocity; derivatives turn "how much" functions into "how fast" functions.
What is the power rule?
For any power of x: d/dx[xⁿ] = n·xⁿ⁻¹ — bring the exponent down as a multiplier and reduce it by one. So d/dx[x³] = 3x², and constants differentiate to zero. Combined with linearity, it differentiates any polynomial.
How do I differentiate more complex functions?
Use the product rule, (fg)′ = f′g + fg′; the quotient rule for ratios; and the chain rule, (f(g(x)))′ = f′(g(x))·g′(x), for compositions. Nearly every function you meet is built from pieces these three rules handle.
What does it mean when the derivative is zero?
The tangent line is horizontal — a critical point. It may be a local maximum, a local minimum, or neither (like x³ at 0); the second derivative or a sign chart tells you which. This is the basis of optimization problems.
Where are derivatives used in real life?
Velocity and acceleration in physics, marginal cost and revenue in economics, rates of reaction in chemistry, and gradient descent — the algorithm that trains neural networks — in machine learning. Any quantity that changes over time has a derivative story.