Derivative Calculator

About the Derivative Calculator

Type a function such as x^2 sin(x), sin(x^2 + 1)/x or sqrt(x^2 + 1) and get its derivative in simplified form, with a tree of the rules used at every step — the power, constant multiple, sum, product, quotient and chain rules, exponentials and logarithms, trigonometric, inverse trigonometric and hyperbolic functions, and logarithmic differentiation for powers such as xˣ. Derivatives of any order up to the 10th are given, each with its own steps.

Other modes find partial and mixed partial derivatives, dy/dx by implicit differentiation, the gradient, the Jacobian matrix with its determinant, and the Hessian matrix with the second-derivative test. Give a point to get exact values such as f′(π) = −π², the tangent and normal lines, and a graph of f with f′ and the tangent. Every result is compared with a numerical derivative wherever floating-point arithmetic can judge it (the page says when it cannot), values at poles such as tan(π/2) are reported as undefined, and nothing leaves your browser.

Common uses

• Checking calculus homework and seeing which rule applies at each step.
• Finding the slope, tangent line and normal line of a curve at a given point.
• Partial derivatives, gradients and Jacobians for multivariable calculus, optimisation and physics.
• Classifying critical points of a function of two or more variables with the Hessian.

The rules used

• Power rule: d/dx[xⁿ] = n·xⁿ⁻¹ (for any real n, so √x and 1/x² are covered by rewriting them as powers).
• Constant multiple and sum rules: d/dx[c·u] = c·u′ and d/dx[u + v] = u′ + v′.
• Product rule: d/dx[uv] = u′v + uv′. Quotient rule: d/dx[u/v] = (u′v − uv′)/v².
• Chain rule: d/dx[f(g(x))] = f′(g(x))·g′(x) — every rule above is combined with it when the inside is not just x.
• Exponentials and logarithms: d/dx[eᵘ] = eᵘ·u′, d/dx[aᵘ] = aᵘ·ln a·u′, d/dx[ln u] = u′/u, d/dx[log_b u] = u′/(u ln b).
• Trigonometry: (sin u)′ = cos u·u′, (cos u)′ = −sin u·u′, (tan u)′ = sec²u·u′; (arcsin u)′ = u′/√(1 − u²), (arctan u)′ = u′/(1 + u²), and the hyperbolic versions.
• Logarithmic differentiation: d/dx[uʷ] = uʷ·(w′ ln u + w·u′/u), used when both the base and the exponent depend on x.

Tangent lines, implicit derivatives and the Hessian

The tangent line at x = a is y − f(a) = f′(a)(x − a); the normal line is perpendicular to it, with slope −1/f′(a) (a vertical line when the tangent is horizontal). For an equation F(x, y) = 0, implicit differentiation treats y as a function of x, so each y-term gets a factor dy/dx by the chain rule; solving gives dy/dx = −Fₓ/F_y. The gradient ∇f collects the first partial derivatives and points in the direction of fastest increase. The Hessian collects the second partial derivatives; at a critical point (∇f = 0) of a function of two variables, D = f_xx·f_yy − f_xy² > 0 with f_xx > 0 means a local minimum, with f_xx < 0 a local maximum, D < 0 a saddle point, and D = 0 tells nothing. With more variables the signs of the leading principal minors decide (Sylvester’s criterion).

Limitations

• Real-valued functions only. Answers are simplified, but an equivalent form may look different from your textbook’s (for example cos²x − sin²x instead of cos 2x).
• Functions defined only at whole numbers (n!, binomial coefficients) and step functions (floor, ceiling, sign) are not differentiated. The derivative of |x| is given as x/|x|, valid for x ≠ 0.
• Orders up to 10, and up to 6 variables and 6 functions for the Jacobian. Very long results are stopped with a message.

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