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Derivative Calculator

Compute derivative formulas with step-by-step differentiation

Select Function Type

Polynomial ax^n Trig sin(ax) Exp e^(ax) Log ln(ax)

f(x) = a*x^n

f(x)=x^

Derivative Rules

Power rule: d/dx(x^n) = n*x^(n-1)

Trig: d/dx sin(x) = cos(x), d/dx cos(x) = -sin(x)

Exp: d/dx e^x = e^x, Log: d/dx ln(x) = 1/x

Chain rule: d/dx f(g(x)) = f(g(x)) * g(x)

The derivative measures the instantaneous rate of change of a function. It is the foundation of calculus, with applications in physics (velocity), economics (marginal analysis), and optimization.

⚠ The derivative at a point equals the slope of the tangent line at that point. If derivative = 0, the point is a critical point.

What Is a Derivative?

The derivative f(x) = lim h->0 (f(x+h)-f(x))/h. It represents the slope of the tangent line. Key rules include power rule, product rule, quotient rule, and chain rule. Each function type has specific differentiation formulas.

Power Rule

d/dx(x^n) = n*x^(n-1). Works for any real n. Example: x^3 -> 3x^2, 1/x = x^(-1) -> -x^(-2) = -1/x^2.

Chain Rule

d/dx f(g(x)) = f(g(x)) * g(x). Differentiate outer, then multiply by inner derivative. Used for all composite functions.

Trig Derivatives

d/dx sin(x)=cos(x), cos(x)->-sin(x), tan(x)->sec^2(x). Chain rule: sin(2x) -> 2*cos(2x).

Exp and Log

d/dx e^x = e^x (unchanged). d/dx ln(x) = 1/x. d/dx a^x = a^x * ln(a). Chain rule: e^(2x) -> 2*e^(2x).

Teaching Example: f(x)=3x^2. Power rule: f(x)=3*2*x^(2-1)=6x^1=6x. Check: at x=2, slope of tangent = 12.

Applications

Physics Optimization Economics Engineering Machine Learning

Frequently Asked Questions

What is derivative?▼

Rate of change. Slope of tangent line. Power rule: d/dx(x^n)=n*x^(n-1). Chain rule for nested functions.

Power rule formula?▼

d/dx(x^n) = n*x^(n-1). Multiply by the exponent, then subtract 1 from the exponent.

Chain rule explained?▼

Differentiate outer, multiply by derivative of inner. f(g(x)) = f(g(x)) * g(x). Each layer adds a factor.

Derivative of e^x?▼

d/dx e^x = e^x. d/dx e^(ax) = a*e^(ax). The exponential function is its own derivative (with chain rule factor).

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