Calxsolver
Calculus

Derivative Calculator

Enter any function of x to get its symbolic derivative — not just polynomials. The calculator applies the chain rule, the product rule, the quotient rule and the standard derivatives of sin, cos, tan, exp, ln and roots, then simplifies the result.

Derivative Calculator

Symbolic derivative of any function — chain, product and quotient rules.

Try:
Answerf'(x) = 2·cos(x²)·x·x + sin(x²)
  1. Functionf(x) = sin(x²)·x
  2. Apply differentiation rulescos(x²)·2·x^(2 − 1)·1·x + sin(x²)·1
  3. Simplify2·cos(x²)·x·x + sin(x²)
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Formula and method

d/dx[f(g(x))] = f′(g(x))·g′(x) [chain rule] d/dx[u·v] = u′v + uv′ [product rule] d/dx[u/v] = (u′v − uv′)/v² [quotient rule]

Applies symbolic differentiation using the chain rule for composite functions, the product and quotient rules for combined terms, and the standard derivatives of sin, cos, tan, exp, ln, sqrt and xⁿ. Functions like xˣ are rewritten as exp(x·ln x) before differentiating. The result is simplified algebraically before display.

Worked examples

Differentiate x³ − 3x Apply the power rule term by term to find the derivative.

Learn why this works

Derivative Rules from First Principles How the limit definition leads to the power, sum, and constant rules → Proof of the Product Rule Why (fg)′ = f′g + fg′ — step-by-step derivation → Proof of the Chain Rule Why (f∘g)′(x) = f′(g(x))·g′(x) — with worked example → Proof of the Quotient Rule Derive (f/g)′ = (f′g − fg′)/g² from the product rule →

Frequently asked questions

What functions are supported?

Any combination of +, −, ×, ÷, ^ and the functions sqrt, sin, cos, tan, exp, ln, log, asin, acos, atan, applied to x.

Does it apply the chain rule?

Yes. Composite functions like sin(x²) and exp(−x²) are differentiated with the chain rule automatically.

What about exponential equations like xˣ?

The calculator rewrites f(x)^g(x) as exp(g·ln f) internally, so it differentiates correctly even when both base and exponent depend on x.

Related solvers

Polynomial Derivative Differentiate a polynomial exactly with the power rule. Polynomial Integral Integrate a polynomial — indefinite or definite. Integral Calculator Symbolic antiderivative — indefinite or definite — with full working. Derivative at a Point Numeric derivative of any function of x at a point. Definite Integral Numeric definite integral of any function of x. Limit Calculator Estimate the limit of a function as x approaches a value. Tangent Line Calculator Equation of the tangent line to a curve at a point.