Product Rule Calculator
Calculate derivatives of products using d/dx[f·g] = f’g + fg’. Differentiate two functions multiplied together with step-by-step solutions.
∫ Calculus📐 d/dx[f·g] = f’g + fg’🔢 Math
f: coefficient a
f: exponent n
g: coefficient b
g: exponent m
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Product Rule Calculator | — | d/dx[f·g] = f’g + fg’ | expression |
Step-by-Step Examples
Example 1
x²·x³
d/dx[x²·x³]
- f'=2x, g=x³; f=x², g'=3x²
- 2x·x³+x²·3x²=5x⁴; verify: d/dx[x⁵]=5x⁴ ✓
✓ 5x⁴
Example 2
3x²·4x³
d/dx[3x²·4x³]
- f'=6x·4x³+3x²·12x²=24x⁴+36x⁴=60x⁴
- Verify: 12x⁵→60x⁴ ✓
✓ 60x⁴
Example 3
Not f'g'
Common error: (fg)'≠f'·g'
- Correct: f'g+fg'=5x⁴
- Wrong: 2x·3x²=6x³
✓ Must use f'g+fg'
Real-World Applications
Physics
KE=½mv² when both m,v depend on time.
Economics
Revenue=price×quantity when both vary.
Engineering
Convolution in signal processing.
ML
Gradient of products of learned parameters.
Common Mistakes to Avoid
⚠️
Multiplying derivatives
(fg)'≠f'·g'. Use f'g+fg'.
⚠️
Missing second term
Both f'g AND fg' are required.
⚠️
Wrong assignment
First term: differentiate f keep g; second: keep f differentiate g.
Frequently Asked Questions
Product rule? ▾
d/dx[f·g]=f'g+fg'. First d-second + second d-first.
Three functions? ▾
d/dx[fgh]=f'gh+fg'h+fgh'. One derivative per term.
Why (fg)'≠f'g'? ▾
Cross terms from the limit definition don't vanish.
Product vs chain rule? ▾
x²·sin(x): product. sin(x²): chain.
Mnemonic? ▾
"First d-second plus second d-first"
Related Math Calculators
Formula Explorer connections
Interpretation: This formula measures limiting behavior, instantaneous change, accumulation or multivariable variation. Assumption: The function must satisfy the continuity or differentiability conditions required by the method. Check domains, bounds, orientation and singularities.