d/dx Derivative Calculator

Calculate derivatives analytically and numerically. Find the derivative formula, evaluate at a point, and understand slope and tangent lines.

Derivative Formula

Select Function

Power n (for xⁿ only)

Derivative f'(x)

Evaluate Derivative at a Point

Function

Point x =

f(x) value

f'(x) value

Tangent Line at a Point

Function

Point x = a

Tangent Line Equation

Differentiation Rules Reference

Power Rule: d/dx [xⁿ] = n·xⁿ⁻¹

Chain Rule: d/dx [f(g(x))] = f'(g(x)) · g'(x)

Product Rule: d/dx [u·v] = u'v + uv'

Quotient Rule: d/dx [u/v] = (u'v − uv') / v²

d/dx [sin x] = cos x

d/dx [cos x] = −sin x

d/dx [eˣ] = eˣ

d/dx [ln x] = 1/x

Numerical Differentiation

When exact formulas aren't available, use: f'(x) ≈ [f(x+h) − f(x−h)] / (2h) where h is a small number like 0.0001.

How the Derivative Calculator Works

How Derivatives Work

A derivative measures the instantaneous rate of change of a function at a specific point. It represents the slope of the tangent line to the function’s graph at that point.

Mathematical Formulas:

Definition (Limit):
f'(x) = lim[h→0] [f(x+h) – f(x)] / h
Fundamental definition of derivative

Power Rule:
If f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹
Most common differentiation rule

Numerical Approximation:
f'(x) ≈ [f(x+h) – f(x-h)] / (2h)
Central difference method

Real-World Applications:

Physics: Velocity (derivative of position), acceleration (derivative of velocity)
Economics: Marginal cost, marginal revenue, elasticity of demand
Engineering: Rate of heat transfer, stress analysis, optimization
Biology: Population growth rates, reaction rates in biochemistry
Finance: Option pricing models, risk analysis, portfolio optimization

Usage Recommendations:

Use analytical method for exact results with simple functions
Choose numerical method for complex functions or when exact form is unknown
For optimization problems, find where derivative equals zero (critical points)
Use second derivative test to determine if critical points are maxima or minima
For numerical methods, use smaller step sizes for higher accuracy