Quotient Rule Calculator
Differentiate a quotient of two functions. Enter numerator and denominator expressions to compute derivative formula and simplified result automatically here.
Equation Preview
Helping Notes
Differentiate f and g separately, then apply (f'·g − f·g') / g². This follows standard definitions of the rule.
Keep expressions consistent in x. Simplification combines like terms automatically for cleaner results.
Results
Inputs Used
Individual Derivatives
Quotient Rule Assembly
Derivative (Unsimplified)
Derivative (Simplified)
What Is a Quotient Rule Calculator?
A Quotient Rule Calculator finds the derivative of a function that is a ratio of differentiable functions, typically written as . The derivative follows the standard rule wherever . This tool computes and symbolically when possible, simplifies the result, and highlights domain restrictions and removable/non‑removable discontinuities. It can also evaluate the slope at a specific point, construct the tangent line, and show a linear approximation for quick estimates. The calculator supports algebraic, trigonometric, exponential, and logarithmic expressions, and it explains each intermediate step so you can verify the reasoning.
About the Quotient Rule Calculator
The quotient rule is a direct consequence of the product rule applied to . If are differentiable at and , then the derivative at exists and equals . When , the quotient is undefined there; the tool reports vertical asymptotes or holes as appropriate. Useful corollaries include the reciprocal rule and logarithmic differentiation to simplify complicated ratios before differentiating. The calculator provides clean, factorized outputs, rationalizes where helpful, and can display alternative equivalent forms (e.g., factoring common terms in the numerator) for clearer interpretation and grading consistency.
Quotient rule:
Point derivative:
Reciprocal: Product link:
Tangent line: with
Domain: (exclusions shown in the final answer)
How to Use This Quotient Rule Calculator
- Enter and , then choose an evaluation point if you need a numeric slope or tangent.
- Compute. The tool differentiates components, assembles , simplifies algebraically, and flags domain restrictions.
- Optional: view the tangent line and linearization .
- Copy the final derivative in your preferred form (factored or expanded) along with the step‑by‑step justification.
Examples
- Polynomial ratio: ⇒ .
- Trig over polynomial: (for ) ⇒ .
- Exponential ratio: ⇒ .
- Tangent at a point: For at , , so .
Formula Snippets Ready for Rendering
FAQs
When should I use the quotient rule?
Use it whenever the function is a ratio and simplifying to a product is inconvenient or obscures domain restrictions.
Can I avoid the quotient rule by rewriting?
Yes. Write and apply the product and chain rules, which yield the same formula.
What happens where g(x) = 0?
The function is undefined; expect vertical asymptotes or holes. The calculator flags such points explicitly.
How do I find a tangent line quickly?
Evaluate and , then use .
Does the rule extend to higher derivatives?
Yes. Differentiate the result again, applying product/quotient rules repeatedly and simplifying after each step.
Are there common algebra mistakes?
Forgetting the order (low d high minus high d low) or omitting in the denominator are the most frequent errors.