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Calculus
University
Calculus (University Level)
Partial Derivative
Calculate partial derivative ∂(x²y)/∂x = 2xy
Standard Formula
∂f/∂x = lim[h→0] (f(x+h,y) - f(x,y))/h
Interactive Sandbox
Live Reactive
Adjust values in real-time and observe mathematical cause-and-effect
x
120
y
120
Live EquationFormula Proof
∂f/∂3 = lim[h→0] (f(3+h,4) - f(3,4))/h = 24
Tangent Line dy/dx
Calculated Solution
24
Direct computational model evaluating Partial Derivative.
Variable Mechanics
Roles, constraints & impacts
2 Params
x
X value
X value
ImpactPrimary formula parameter
ConstraintReal number
y
Y value
Y value
ImpactPrimary formula parameter
ConstraintReal number
At a Glance
Gotchas, units & real-world use
Essential
Formula Nature & Mechanics
Direct computational model evaluating Partial Derivative.
Common Mistakes & Traps
- Ensure all input measurements share consistent units before calculating.
- Check for denominator values approaching zero to prevent undefined division.
- Follow standard order of operations (PEMDAS: Parentheses, Exponents, Multiply/Divide, Add/Subtract).
Dimensional Analysis & Units
Direct evaluation matching formula variable dimensions.
Real-World & Industry Application
Widely utilized across modern calculus applications, academic analysis, and engineering estimations.
Worked Example: Partial Derivative
Given:
- x = 3
- y = 4
- ∂(x²y)/∂x = 2xy
- At (3,4): 2(3)(4) = 24
Answer: 24
Frequently Asked Questions
What is the Partial Derivative formula?
∂f/∂x = lim[h→0] (f(x+h,y) - f(x,y))/h. Calculate partial derivative ∂(x²y)/∂x = 2xy.
How do I calculate Partial Derivative?
Enter X value (x), Y value (y) into the calculator. It applies ∂f/∂x = lim[h→0] (f(x+h,y) - f(x,y))/h and shows every step of the working.
Related Calculus Calculators
Power Rule for Derivativesd/dx(xⁿ) = n × x^(n-1)Definite Integral (Power Rule)∫[a to b] x^n dx = [x^(n+1)/(n+1)]|[a to b]Chain Rule (Derivative)d/dx[f(g(x))] = f'(g(x)) · g'(x)Product Rule (Derivative)d/dx[f·g] = f'·g + f·g'Quotient Rule (Derivative)d/dx[f/g] = (f'·g − f·g') / g²Derivative of sin(ax)d/dx[sin(ax)] = a·cos(ax)
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