Derivative Calculator

Compute symbolic derivatives of any function using the variable x. Supports 1st, 2nd, and 3rd order derivatives. Can evaluate the derivative at a specific x value. Uses the mathjs symbolic differentiation engine.

Function f(x)

Quick examples:

Derivative order

Evaluate at x = (optional)

Differentiation Rules

Power rule: d/dx(xⁿ) = n·xⁿ⁻¹ Product rule: d/dx(f·g) = f′g + fg′ Quotient rule: d/dx(f/g) = (f′g − fg′) / g² Chain rule: d/dx(f(g(x))) = f′(g(x))·g′(x) sin(x) → cos(x) cos(x) → −sin(x) eˣ → eˣ ln(x) → 1/x log₁₀(x) → 1/(x·ln10) √x → 1/(2√x)

Geometric Interpretation

f′(x) > 0	Function is increasing at x
f′(x) < 0	Function is decreasing at x
f′(x) = 0	Critical point (potential max/min)
f″(x) > 0	Concave up (cup shape)
f″(x) < 0	Concave down (cap shape)
f″(x) = 0	Possible inflection point

How to use

Supported functions

All standard functions are supported: sin, cos, tan, asin, acos, atan, exp, log (natural), log10, sqrt, abs, as well as polynomial expressions, products, quotients, and chains.

Notation

Use * for multiplication (2*x not 2x), ^ for powers (x^3), and parentheses for grouping. Constants: pi = π, e = 2.718…

Evaluate at a point

Enter an x value in the optional field to compute the numeric value of the derivative there — useful for finding the slope of a tangent line at a specific point.

Higher-order derivatives

Select f″ for the second derivative (rate of change of rate of change, used in concavity analysis) or f‴ for the third.

Formula

Power rule: d/dx(xⁿ) = n·xⁿ⁻¹ Product rule: d/dx(fg) = f′g + fg′ Chain rule: d/dx(f(g(x))) = f′(g(x))·g′(x) sin → cos, cos → −sin, eˣ → eˣ, ln(x) → 1/x

Tips

•The derivative at a point equals the slope of the tangent line to the curve at that point.
•Where f′(x) = 0 are the critical points (local maxima/minima). Where f″(x) = 0 are inflection points.
•Use the graphing calculator to visualise f(x) and f′(x) together.