Product Rule (Derivative)
Differentiate a product of two power functions: f(x) = xᵃ and g(x) = xᵇ at a given x.
Interactive Sandbox
Adjust values in real-time and observe mathematical cause-and-effect
Direct computational model evaluating Product Rule (Derivative).
Variable Mechanics
Roles, constraints & impacts
Exponent of f(x)
Exponent of f(x)
Exponent of g(x)
Exponent of g(x)
Value of x
Value of x
At a Glance
Gotchas, units & real-world use
Formula Nature & Mechanics
Direct computational model evaluating Product Rule (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: Product Rule (Derivative)
Given:
- a = 2
- b = 3
- x = 1
- f=x², g=x³
- f'=2x, g'=3x²
- d/dx = 2x·x³ + x²·3x² = 2x⁴+3x⁴ = 5x⁴
- At x=1: 5
Answer: 5
Frequently Asked Questions
What is the Product Rule (Derivative) formula?
d/dx[f·g] = f'·g + f·g'. Differentiate a product of two power functions: f(x) = xᵃ and g(x) = xᵇ at a given x.
How do I calculate Product Rule (Derivative)?
Enter Exponent of f(x) (a), Exponent of g(x) (b), Value of x (x) into the calculator. It applies d/dx[f·g] = f'·g + f·g' and shows every step of the working.
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