Partial Derivative Calculator – Compute ∂/∂x or ∂/∂y and Evaluate at a Point

Introduction

The discipline of calculus requires working with multivariable functions, as do much of optimization, machine learning, physics and economics. When your function involves more variables than one–f(x, y), say–you frequently require the rate of change with respect to a certain variable in a certain value of the other variable. A partial derivative explicitly provides that to you.

The Partial Derivative Calculator on AizoroTools.com will do it easy and painless. Type in a function f(x, y) and choose whether differentiation with respect to x or y is to be specified, and type in (optionally) a point (x0,y0)(x_0, y_0)(x0​,y0​) at which to compute the derivative numerically. The tool gives the symbolic partial derivative on the fly, and, when a point is specified, the numeric value of the partial derivative at this point.

You can check homework, verify calculations in a proof, and debug a gradient-based optimization, in all of these situations using this calculator saves you time and minimizes mistakes.

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Inputs Required for the Calculator

Your calculator accepts these simple inputs:

1) Function f(x, y)

Type your function using standard math notation and operators:

• Polynomials: x^2 + 3xy - y^3

• Products/quotients: (sin(x)*y) / (x^2 + y^2)

• Exponentials & logs: e^(x*y), ln(x + y)

• Trig functions: sin(x), cos(y), tan(x*y)

• Other: sqrt(x^2 + y^2), abs(x - y) (where differentiable)

Tip: Use ^ for powers, * for multiplication, exp(...) as e^(...) if you prefer, and ln for natural log.

2) Differentiate with respect to

Choose x or y from the dropdown.

• ∂f/∂x: vary x and hold y constant

• ∂f/∂y: vary y and hold x constant

3) Evaluate at (optional)

Provide a point (x0,y0)(x_0, y_0)(x0​,y0​) to compute a numeric value:

• Example: Evaluate at $x=1$, $y=2$

If you leave this blank, the calculator returns only the symbolic partial derivative.

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What the Calculator Outputs

• Symbolic partial derivative (e.g., ∂∂xf(x,y)\frac{\partial}{\partial x} f(x, y)∂x∂​f(x,y) or ∂∂yf(x,y)\frac{\partial}{\partial y} f(x, y)∂y∂​f(x,y))

• Simplified expression (whenever possible)

• Numeric evaluation at a point (x0,y0)(x_0, y_0)(x0​,y0​) if provided

• Error and domain checks (e.g., division by zero, log of a nonpositive number)

• (Optional) Gradient hint: If you compute both ∂/∂x and ∂/∂y, you have the gradient ∇f=⟨fx,fy⟩\nabla f = \langle f_x, f_y \rangle∇f=⟨fx​,fy​⟩

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How the Partial Derivative Calculator Works

1. Parse your input function and normalize notation (powers, functions, parentheses).

2. Differentiate symbolically with respect to your chosen variable using standard rules (linearity, product/quotient rules, chain rule, trig/exponential/log derivatives).

3. Simplify the resulting expression (common factors, trig identities when applicable).

4. Evaluate at a point if values (x0,y0)(x_0, y_0)(x0​,y0​) were provided and the expression is defined there.

5. Return results with clear symbolic and numeric outputs.

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Refresher: Partial Derivative Rules (Quick Guide)

For a function $f(x,y)$:

• Linearity
  ∂∂x[a u+b v]=a ∂u∂x+b ∂v∂x\frac{\partial}{\partial x}[a\,u + b\,v] = a\,\frac{\partial u}{\partial x} + b\,\frac{\partial v}{\partial x}∂x∂​[au+bv]=a∂x∂u​+b∂x∂v​

• Product rule
  ∂∂x[u v]=ux v+u vx\frac{\partial}{\partial x}[u\,v] = u_x\,v + u\,v_x∂x∂​[uv]=ux​v+uvx​

• Quotient rule
  ∂∂x ⁣[uv]=ux v−u vxv2\frac{\partial}{\partial x}\!\left[\frac{u}{v}\right] = \frac{u_x\,v – u\,v_x}{v^2}∂x∂​[vu​]=v2ux​v−uvx​​ (where v≠0v \neq 0v=0)

• Chain rule (single inner)
  If $u=g(x,y)$ and $f=h(u)$, then
  ∂f∂x=h′(u)⋅∂g∂x\frac{\partial f}{\partial x} = h'(u) \cdot \frac{\partial g}{\partial x}∂x∂f​=h′(u)⋅∂x∂g​

• Chain rule (composition like $xy$)
  For $f(x,y)=\sin (xy)$:
  ∂f∂x=cos⁡(xy)⋅y\frac{\partial f}{\partial x} = \cos(xy)\cdot y∂x∂f​=cos(xy)⋅y,
  ∂f∂y=cos⁡(xy)⋅x\frac{\partial f}{\partial y} = \cos(xy)\cdot x∂y∂f​=cos(xy)⋅x

• Common derivatives
  ∂∂x[xn]=n xn−1\frac{\partial}{\partial x}[x^n] = n\,x^{n-1}∂x∂​[xn]=nxn−1, ∂∂x[y]=0\frac{\partial}{\partial x}[y] = 0∂x∂​[y]=0
  ∂∂x[sin⁡x]=cos⁡x\frac{\partial}{\partial x}[\sin x] = \cos x∂x∂​[sinx]=cosx, ∂∂x[cos⁡x]=−sin⁡x\frac{\partial}{\partial x}[\cos x] = -\sin x∂x∂​[cosx]=−sinx
  ∂∂x[ex]=ex\frac{\partial}{\partial x}[e^{x}] = e^{x}∂x∂​[ex]=ex, ∂∂x[ln⁡x]=1/x\frac{\partial}{\partial x}[\ln x] = 1/x∂x∂​[lnx]=1/x (domain permitting)

In partial derivatives, the other variable is treated as a constant.

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Example Calculations

Example 1 — Polynomial + Product

Input: f(x,y)=x2y+3xy2f(x, y) = x^2y + 3xy^2f(x,y)=x2y+3xy2
Differentiate w.r.t. x:

∂f∂x=2x y+3y2\frac{\partial f}{\partial x} = 2x\,y + 3y^2∂x∂f​=2xy+3y2

Evaluate at (x, y) = (1, 2):
2(1)(2)+3(22)=4+12=162(1)(2) + 3(2^2) = 4 + 12 = 162(1)(2)+3(22)=4+12=16

Output:

• Symbolic: 2xy+3y22xy + 3y^22xy+3y2

• Numeric at (1, 2): 16

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Example 2 — Trig + Chain Rule

Input: $f(x,y)=\sin (xy)+y\cos x$
Differentiate w.r.t. x:

• For $\sin (xy)$: derivative is $\cos (xy)\cdot y$

• For $y\cos x$: derivative is $y(-\sin x)$

∂f∂x=ycos⁡(xy)−ysin⁡x\frac{\partial f}{\partial x} = y\cos(xy) – y\sin x∂x∂f​=ycos(xy)−ysinx

Evaluate at (x, y) = (\pi, 1):
$1\cdot \cos (\pi )-1\cdot \sin (\pi )=(-1)-0=-1$

Output:

• Symbolic: $y\cos (xy)-y\sin x$

• Numeric at $(\pi ,1)$: −1

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Example 3 — Exponential & Log (with domain)

Input: f(x,y)=ex+y ln⁡(x−y)f(x, y) = e^{x+y}\,\ln(x – y)f(x,y)=ex+yln(x−y) (requires $x-y>0$)
Differentiate w.r.t. y:

Use product rule: u=ex+yu = e^{x+y}u=ex+y, $v=\ln (x-y)$

• uy=ex+yu_y = e^{x+y}uy​=ex+y

• vy=1x−y⋅(−1)=−1x−yv_y = \frac{1}{x – y}\cdot(-1) = -\frac{1}{x – y}vy​=x−y1​⋅(−1)=−x−y1​

∂f∂y=uy⋅v+u⋅vy=ex+yln⁡(x−y)−ex+y⋅1x−y=ex+y[ln⁡(x−y)−1x−y]\frac{\partial f}{\partial y} = u_y \cdot v + u\cdot v_y = e^{x+y}\ln(x – y) – e^{x+y}\cdot\frac{1}{x – y} = e^{x+y}\left[\ln(x – y) – \frac{1}{x – y}\right]∂y∂f​=uy​⋅v+u⋅vy​=ex+yln(x−y)−ex+y⋅x−y1​=ex+y[ln(x−y)−x−y1​]

Evaluate at (x, y) = (3, 1): Domain OK ($3-1=2>0$).

e4[ln⁡(2)−12]e^{4}\left[\ln(2) – \tfrac{1}{2}\right]e4[ln(2)−21​]

The calculator will give a numeric approximation.

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Example 4 — Quotient Rule

Input: f(x,y)=x2+yx+y2f(x, y) = \dfrac{x^2 + y}{x + y^2}f(x,y)=x+y2x2+y​ with x+y2≠0x + y^2 \neq 0x+y2=0
Differentiate w.r.t. x:

Let u=x2+yu = x^2 + yu=x2+y, v=x+y2v = x + y^2v=x+y2.
ux=2x, vx=1u_x = 2x,\ v_x = 1ux​=2x, vx​=1

∂f∂x=uxv−uvxv2=(2x)(x+y2)−(x2+y)(1)(x+y2)2=2×2+2xy2−x2−y(x+y2)2=x2+2xy2−y(x+y2)2\frac{\partial f}{\partial x} = \frac{u_x v – u v_x}{v^2} = \frac{(2x)(x + y^2) – (x^2 + y)(1)}{(x + y^2)^2} = \frac{2x^2 + 2xy^2 – x^2 – y}{(x + y^2)^2} = \frac{x^2 + 2xy^2 – y}{(x + y^2)^2}∂x∂f​=v2ux​v−uvx​​=(x+y2)2(2x)(x+y2)−(x2+y)(1)​=(x+y2)22x2+2xy2−x2−y​=(x+y2)2x2+2xy2−y​

Provide a point to evaluate numerically if needed.

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Why Use This Calculator?

• ✅ Instant symbolic derivatives — no manual rule juggling

• ✅ Accurate evaluation at specific points

• ✅ Helpful error checks for domain restrictions

• ✅ Clear steps to learn while you compute

• ✅ Great for study & research: gradients, tangents, and sensitivity analysis

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Practical Applications

• Optimization & ML: partials build the gradient used in gradient descent.

• Economics: marginal rates (e.g., holding price fixed while changing quantity).

• Physics/Engineering: flux, fields, and potentials often depend on multiple variables.

• Statistics: log-likelihood functions with multiple parameters.

• Graphics & Geometry: slope of surfaces $z=f(x,y)$ via fxf_xfx​ and fyf_yfy​.

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How to Use the Partial Derivative Calculator

1. Open the calculator on AizoroTools.com.

2. Enter your function f(x, y) exactly as you’d write it in class.

3. Choose w.r.t. x or w.r.t. y.

4. (Optional) Enter a point (x0,y0)(x_0, y_0)(x0​,y0​) to evaluate numerically.

5. Click Calculate to see the symbolic derivative and (if applicable) the numeric value.

6. If you need the other partial, switch the dropdown and recalculate.

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Tips & Best Practices

• Parentheses are your friend: (x^2 + y)/(x + y^2) is safer than x^2 + y/x + y^2.

• Check domains: logs need positive arguments, denominators can’t be zero, square roots require nonnegative radicands.

• Simplify before evaluating: The tool attempts this automatically; cleaner forms reduce numeric error.

• Compute both partials for gradients: ∇f=(fx,fy)\nabla f = (f_x, f_y)∇f=(fx​,fy​) is crucial for optimization and tangent-plane analysis.

• Test a few points: If your function’s behavior changes by region, evaluating at multiple points can reveal more.

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Frequently Asked Questions (FAQs)

Q1: What’s the difference between dfdx\frac{df}{dx}dxdf​ and ∂f∂x\frac{\partial f}{\partial x}∂x∂f​?
A: dfdx\frac{df}{dx}dxdf​ is a total derivative (single-variable or accounts for how all variables change with $x$), while ∂f∂x\frac{\partial f}{\partial x}∂x∂f​ is a partial derivative, holding other variables constant.

Q2: Can I include absolute values, piecewise parts, or non-smooth functions?
A: Yes, but derivatives may not exist at non-smooth points (e.g., at $x=0$ for $|x|$). The calculator will return piecewise indications or warnings where possible.

Q3: Will it handle complicated compositions like ln⁡(sin⁡(xy)2+ex−y)\ln(\sin(xy)^2 + e^{x-y})ln(sin(xy)2+ex−y)?
A: Yes—just enter it carefully with parentheses. The tool applies the multivariable chain rule under the hood.

Q4: Can I get both fxf_xfx​ and fyf_yfy​ at once?
A: Compute one, then toggle the variable and compute the other. Together they form the gradient.

Q5: What if I get an undefined result at my evaluation point?
A: That usually means a domain issue (e.g., division by zero or invalid log argument). Adjust your point or check the function’s domain.

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Conclusion

With AizoroTools.com Partial Derivative Calculator, you can differentiate common matrices. Enter f(x, y), simplify w.r.t. x or y, and evaluate at a point to get immediate symbolic and numeric output–with a domain check and uncomplicated simplification.

Whether checking homework, studying to take an exam, or doing modeling in the real world, having the clarity and accuracy you need to do it quickly, easily is what this tool provides.

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