Derivative Calculator
Calculate derivatives of common functions step by step.
Enter a function type and coefficients to find the derivative using standard differentiation rules.
Derivatives measure the instantaneous rate of change of a function — how fast the output changes as the input changes. Visually, the derivative at any point equals the slope of the tangent line to the curve at that point.
Formal definition (limit definition): f′(x) = lim[h→0] (f(x+h) − f(x)) / h
In practice, we use differentiation rules to find derivatives without computing limits every time.
Core differentiation rules:
| Rule | Formula | Example |
|---|---|---|
| Power Rule | d/dx [xⁿ] = n·xⁿ⁻¹ | d/dx [x³] = 3x² |
| Constant | d/dx [c] = 0 | d/dx [7] = 0 |
| Constant Multiple | d/dx [c·f(x)] = c·f′(x) | d/dx [5x²] = 10x |
| Sum/Difference | d/dx [f±g] = f′±g′ | d/dx [x³+x] = 3x²+1 |
| Product Rule | d/dx [fg] = f′g + fg′ | d/dx [x·sin x] = sin x + x·cos x |
| Chain Rule | d/dx [f(g(x))] = f′(g(x))·g′(x) | d/dx [sin(x²)] = 2x·cos(x²) |
| Quotient Rule | d/dx [f/g] = (f′g − fg′)/g² | — |
Common derivatives:
- d/dx [sin x] = cos x
- d/dx [cos x] = −sin x
- d/dx [eˣ] = eˣ
- d/dx [ln x] = 1/x
- d/dx [tan x] = sec²x
Worked example (chain rule): Find the derivative of f(x) = (3x² + 1)⁴.
- Outer function: u⁴ → derivative 4u³
- Inner function: 3x² + 1 → derivative 6x
- Chain rule: f′(x) = 4(3x² + 1)³ · 6x = 24x(3x² + 1)³
Real-world use: Derivatives power physics (velocity = dx/dt), economics (marginal cost), machine learning (gradient descent), and engineering (optimization). The derivative of position is velocity; the derivative of velocity is acceleration.