Multivariable derivatives in practice
Calculus with several inputs uses partial derivatives: differentiate with respect to one letter while treating the others as constants. A multivariable derivative calculator automates that bookkeeping and shows which power, product, quotient, or chain rule fired — the same skills graded on paper exams.
Gradients, directional derivatives, tangent planes, and many economics or physics models start from these partials. Use this page when your homework is framed as f(x, y) or f(x, y, z) rather than a single-variable y = f(x).
How to use this multivariable tool
- Enter an expression with two or more variables.
- Choose the differentiation variable (x, y, z, …).
- Press Calculate for that partial with free steps.
- Switch the variable and repeat to collect every gradient component.
Looking for a shorter partial-focused explainer? See the dedicated partial derivative calculator. For rates in a chosen direction after you have ∇f, continue to the directional derivative calculator.
Gradient workflow examples
Two variables
For f = x^2*y + y^3, ∂f/∂x = 2xy and ∂f/∂y = x² + 3y², so
∇f = ⟨2xy, x² + 3y²⟩. Run both partials above and assemble the vector in your notebook.
Three variables
For f = x*y*z, each partial freezes two letters:
fₓ = yz, fᵧ = xz, f_z = xy. Three calculator runs build the full gradient.
Compositions
Try sin(x*y) + z or ln(x^2 + y^2). Chain-rule factors appear
inside the steps when the outer function wraps a product or sum of several variables.
When to stay single-variable instead
If every other symbol is a fixed parameter and only x changes, you can still use partial mode — or switch to the ordinary derivative calculator when the problem is truly one-dimensional. Mixed partials and Hessian entries need multiple passes; Clairaut’s theorem checks are easier when each first partial’s steps are already documented.
Implicit surfaces F(x, y) = 0 that ask for dy/dx are a special case of multivariable thinking — use the implicit derivative calculator workflow with Fₓ and Fᵧ.
Input tips for several letters
Write products with * (x*y*z), and set the variable field
deliberately each time you change components. Start at order 1 unless you need a second
partial in the same variable for a Hessian diagonal term.
Multivariable derivative calculator FAQ
What is a multivariable derivative calculator?
A tool that computes derivatives for functions of several variables — typically partials that hold other inputs fixed.
How do I build a gradient here?
Compute each partial by changing the variable between runs, then write ⟨fₓ, fᵧ, …⟩ yourself from the results.
Is this different from the partial page?
Same free engine. This URL targets “multivariable” searches and gradient / 3D homework framing; the partial page is the shorter ∂f/∂x landing.
Is it free?
Yes — partials and steps with no account or Pro subscription.