Quotient Rule Calculator
Calculate derivatives of fractions using d/dx[f/g] = (f’g - fg’) / g². Find derivatives of rational functions step by step.
∫ Calculus📐 d/dx[f/g] = (f’g - fg’) / g²🔢 Math
Numerator: coefficient a
Numerator: exponent n
Denominator: coefficient b
Denominator: exponent m
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Quotient Rule Calculator | — | d/dx[f/g] = (f’g - fg’) / g² | expression |
Step-by-Step Examples
Example 1
x³/x²
d/dx[x³/x²]
- f'=3x²,g=x²; f=x³,g'=2x
- (3x²·x²-x³·2x)/x⁴=(3x⁴-2x⁴)/x⁴=1
- Verify: x³/x²=x, d/dx[x]=1 ✓
✓ 1
Example 2
tan(x)
d/dx[sin/cos]
- (cos·cos-sin·(-sin))/cos²=(cos²+sin²)/cos²=sec²x
✓ sec²(x)
Example 3
2x²/3x
d/dx[2x²/3x]
- (4x·3x-2x²·3)/9x²=6x²/9x²=2/3
- Verify: 2x²/3x=2x/3, d=2/3 ✓
✓ 2/3
Real-World Applications
Physics
Derivatives of density, pressure ratios.
Economics
Marginal average cost = d/dx[TC/q].
Engineering
Transfer functions in control systems.
Statistics
Derivatives of likelihood ratio statistics.
Common Mistakes to Avoid
⚠️
Wrong order
f'g MINUS fg'. Swapping gives wrong sign.
⚠️
Missing g²
Denominator is g-squared, not g.
⚠️
Simplify first
d/dx[x⁵/x²]=d/dx[x³]=3x² is easier.
Frequently Asked Questions
Quotient rule formula? ▾
(f'g-fg')/g². Mnemonic: low d-high minus high d-low over low squared.
Use product rule instead? ▾
f/g=f·g⁻¹. Apply product rule with chain rule on g⁻¹.
d/dx[1/x]? ▾
-1/x² via quotient rule or power rule (x⁻¹).
d/dx[tan(x)]? ▾
=sec²(x).
d/dx[cot(x)]? ▾
-csc²(x).
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.