Chain Rule Partial Derivative Calculator

∂f/∂u (Outer Derivative):

∂u/∂x (Inner Derivative w.r.t x):

∂u/∂y (Inner Derivative w.r.t y):

Understanding how a small change in one variable propagates through nested functions is central to multivariable calculus, physics, machine learning, and engineering. The Chain Rule Partial Derivative Calculator is a tool that automates the algebraic steps for computing partial derivatives when functions depend on other functions — for example, when z=f(x,y)z=f(x,y)z=f(x,y) but xxx and yyy themselves depend on other variables.

This article explains what the calculator does, how the multivariable chain rule works, how to use the tool, a clear worked example, practical tips, common pitfalls, and 20 frequently asked questions to help you get the most out of it.

What this calculator does (simple explanation)

The calculator accepts a multivariable function and the dependency relationships between variables, then:

Computes partial derivatives using the multivariable chain rule,
Expands derivatives into step-by-step terms (so you can follow the logic),
Handles nested dependencies (e.g., z=f(x,y)z=f(x,y)z=f(x,y), x=g(u,v)x=g(u,v)x=g(u,v), y=h(u,v)y=h(u,v)y=h(u,v) or x=g(t)x=g(t)x=g(t), y=h(t)y=h(t)y=h(t)),
Simplifies final expressions where possible (algebraic simplification).

It’s ideal for students, instructors, engineers, data scientists, and anyone who needs reliable, readable derivative computations.

Quick primer — the multivariable chain rule

There are a few common chain-rule patterns. Two typical forms:

1. Single-parameter dependence (one intermediate variable):
If z=f(x,y)z=f(x,y)z=f(x,y) and x=x(t)x=x(t)x=x(t), y=y(t)y=y(t)y=y(t), then the derivative of zzz with respect to ttt is dzdt=∂f∂xdxdt+∂f∂ydydt.\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}.dtdz=∂x∂fdtdx+∂y∂fdtdy.

2. Two intermediate variables (each depends on two new variables):
If z=f(x,y)z=f(x,y)z=f(x,y), x=x(u,v)x=x(u,v)x=x(u,v), y=y(u,v)y=y(u,v)y=y(u,v), then the partial derivative of zzz with respect to uuu is ∂z∂u=∂f∂x∂x∂u+∂f∂y∂y∂u.\frac{\partial z}{\partial u} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial u}.∂u∂z=∂x∂f∂u∂x+∂y∂f∂u∂y.

A similar expression holds for ∂z/∂v\partial z/\partial v∂z/∂v.

The calculator uses these identities and extends to nested compositions of arbitrary depth.

How to use the Chain Rule Partial Derivative Calculator

Enter the main function z=f(x,y,… )z = f(x,y,\dots)z=f(x,y,…). For example: z = x^2 * sin(y) + e^(xy).
Define dependencies — tell the calculator how each argument depends on other variables. Examples:
- x = t^2 + 1 and y = sin(3t) (single parameter t),
- or x = u*v and y = u^2 + v (two parameters u and v).
Choose what derivative you want: dzdt\dfrac{dz}{dt}dtdz, ∂z∂u\dfrac{\partial z}{\partial u}∂u∂z, or a gradient vector.
Compute — the tool shows:
- the partial derivatives ∂f/∂x\partial f/\partial x∂f/∂x, ∂f/∂y\partial f/\partial y∂f/∂y,
- the derivatives of the inner functions (e.g., dx/dtdx/dtdx/dt),
- the assembled chain-rule expression,
- an optionally simplified final expression.
View step-by-step — expand intermediate steps to learn the reasoning or copy the final expression for use in homework, reports, or code.

Worked example (step-by-step)

Problem:
Let z=f(x,y)=x2sin⁡(y)+exyz = f(x,y) = x^2 \sin(y) + e^{xy}z=f(x,y)=x2sin(y)+exy, where x=t2x = t^2x=t2 and y=3ty = 3ty=3t. Compute dzdt\dfrac{dz}{dt}dtdz.

Step 1 — partials of f: ∂f∂x=2xsin⁡(y)+yexy,∂f∂y=x2cos⁡(y)+xexy.\frac{\partial f}{\partial x} = 2x\sin(y) + y e^{xy}, \qquad \frac{\partial f}{\partial y} = x^2 \cos(y) + x e^{xy}.∂x∂f=2xsin(y)+yexy,∂y∂f=x2cos(y)+xexy.

Step 2 — derivatives of inner functions: dxdt=2t,dydt=3.\frac{dx}{dt} = 2t, \qquad \frac{dy}{dt} = 3.dtdx=2t,dtdy=3.

Step 3 — apply the chain rule: dzdt=(2xsin⁡(y)+yexy)⋅(2t)+(x2cos⁡(y)+xexy)⋅3.\frac{dz}{dt} = \left(2x\sin(y) + y e^{xy}\right)\cdot(2t) + \left(x^2 \cos(y) + x e^{xy}\right)\cdot 3.dtdz=(2xsin(y)+yexy)⋅(2t)+(x2cos(y)+xexy)⋅3.

Step 4 — substitute x=t2x=t^2x=t2, y=3ty=3ty=3t: dzdt=(2t2sin⁡(3t)+3te3t⋅t2)⋅2t+(t4cos⁡(3t)+t2e3t3)⋅3.\frac{dz}{dt} = \big(2t^2\sin(3t) + 3t e^{3t\cdot t^2}\big)\cdot 2t + \big(t^4 \cos(3t) + t^2 e^{3t^3}\big)\cdot 3.dtdz=(2t2sin(3t)+3te3t⋅t2)⋅2t+(t4cos(3t)+t2e3t3)⋅3.

The calculator computes each symbolic step and can simplify or leave it expanded depending on user preference.

Practical tips and best practices

Use parentheses when entering expressions (e.g., e^(x*y)).
Specify variables clearly — name the dependent and independent variables.
Check units — when working in physics, keep units consistent before differentiation.
Use decimal evaluation if you need a numeric derivative at a point (e.g., t=1.2t=1.2t=1.2).
Compare symbolic and numeric results to catch algebra mistakes.

Common use cases

Solving physics problems with composite position/velocity relationships,
Backpropagation sanity checks in neural networks (chain rule everywhere!),
Multivariable optimization and sensitivity analysis,
Engineering: stress/strain models where one parameter depends on another,
Education: step-by-step learning for students.

20 Frequently Asked Questions (FAQs)

What kinds of functions can the calculator handle?
Polynomial, trigonometric, exponential, logarithmic, and many composed symbolic expressions.
Can it compute higher-order derivatives?
Yes — you can request second derivatives like d2zdt2\dfrac{d^2z}{dt^2}dt2d2z if inner derivatives are provided.
Does it return symbolic expressions or numeric values?
Both: symbolic steps and optional numeric evaluation at specified points.
Can it handle more than two layers of nesting?
Yes — nested dependencies (e.g., x(u(v))x(u(v))x(u(v))) are supported.
Is vector-valued fff supported?
Basic scalar-output functions are primary; check the tool for vector-output support (Jacobian).
Does it simplify final expressions?
The calculator attempts simplification but leaves complex expressions expanded if needed.
Can I copy the step-by-step output?
Yes — the steps are presented in readable text for copying.
What if my inner functions contain absolute values or piecewise definitions?
Piecewise and nondifferentiable points require special handling; numeric checks are safer.
Can it detect non-differentiable points?
It flags obvious issues (e.g., division by zero, absolute corners) but not every singularity.
Is there support for partial derivatives with respect to multiple variables at once?
Yes — you can request the gradient (∂z/∂u,∂z/∂v)(\partial z/\partial u, \partial z/\partial v)(∂z/∂u,∂z/∂v).
Does the calculator show intermediate substitutions?
Yes — it can show substitution of inner functions into outer partials.
Can I upload my function or do I type it?
Usually you type or paste the function; some tools accept file inputs.
Does it support implicit differentiation?
The tool focuses on explicit chain-rule problems; implicit differentiation may be limited.
Can it export results as LaTeX?
Many calculators offer LaTeX output for inclusion in reports.
Is it useful for machine learning gradient checks?
Absolutely — it helps verify hand-derived gradients for small models.
What if the inner variables depend on each other?
Provide the full dependency graph; the tool will traverse and apply the chain rule accordingly.
Does it handle partial derivatives for functions of more than two variables?
Yes — supply all variable dependencies and request the needed partials.
Are numeric derivatives (finite differences) available?
Numeric derivative options are often included for verification.
Will it simplify expressions like exye^{xy}exy vs ex⋅ye^{x\cdot y}ex⋅y?
The calculator understands both notations if entered clearly.
Is the tool appropriate for exams/homework?
Use it to check work and learn; don’t rely on it if exams restrict calculator use.

Final notes

The Chain Rule Partial Derivative Calculator is more than a shortcut — it’s a learning aid. It reduces algebraic drudgery, highlights each derivative step, and helps you focus on concept and interpretation. Whether you need purely symbolic answers, numeric evaluations at specific points, or stepwise breakdowns for learning and teaching, this tool speeds up correct results and deepens understanding.