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Differentiation methods

How to Use the Quotient Rule and Keep the Order

The quotient rule differentiates a changing ratio. Its subtraction order and squared denominator record how the changing divisor affects the output.

Direct answer

(f/g)′ = (f′g − fg′)/g²

The denominator g(x) must be nonzero.

When to use how to use the quotient rule

Choose the method from the top-level mathematical structure before manipulating symbols. Conditions are part of the task, not optional notes.

Ask first 1

Which structure controls the method?

What to do

Use the quotient rule when a changing expression divides another and simplification does not remove the quotient.

Ask first 2

Which condition can change the answer?

What to do

The denominator g(x) must be nonzero.

Ask first 3

How should the result be checked?

What to do

Verify the squared denominator and compare with an algebraically simplified derivative when possible.

Derive the rule from a reciprocal

Write f/g as f·g⁻¹.

Apply the product rule and the chain rule to g⁻¹.

Combine f′g⁻¹−fg⁻²g′ over g².

Simplify first when valid cancellation removes the quotient, but retain excluded points.

1
fg=f g−1\frac{f}{g} = f\,g^{-1}
2
(g⁻¹)'=−g⁻²g'
3
(f/g)'=(f'g−fg')/g²

Worked examples

ddx[x2+1x]\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{x^{2} + 1}{x}\right]
1−1x21 - \frac{1}{x^{2}}

Valid where x≠0.

ddx[sin⁡xx]\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{\sin x}{x}\right]
[x cos(x)−sin(x)]/x²

Preserve the numerator order.

ddx[xx+1]\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{x}{x + 1}\right]
1(x+1)2\frac{1}{\left(x + 1\right)^{2}}

The numerator simplifies after subtraction.

Common mistakes

  • • Reversing the subtraction.
  • • Forgetting g².
  • • Dividing f′ by g′.
  • • Ignoring denominator zeros.

Error clinic: locate the first invalid step

Incorrect attempt

[sin(x)/x]'=cos(x)

Why it fails

The denominator was treated as constant.

Correction

[x cos(x)−sin(x)]/x²

Incorrect attempt

[x/(x+1)]'=−1/(x+1)²

Why it fails

The subtraction order was reversed.

Correction

1(x+1)2\frac{1}{\left(x + 1\right)^{2}}

Practice the same idea with a changed structure

Name the rule and its conditions before revealing the reference answer. Explain every sign and factor.

ddx[sin⁡xx]\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{\sin x}{x}\right]

Check before solving

Preserve the numerator order.

Reveal the reference answer

Reference answer

[x cos(x)−sin(x)]/x²
ddx[xx+1]\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{x}{x + 1}\right]

Check before solving

The numerator simplifies after subtraction.

Reveal the reference answer

Reference answer

1(x+1)2\frac{1}{\left(x + 1\right)^{2}}

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

微分:sin⁡xx\frac{\sin x}{x}微分:xx+1\frac{x}{x + 1}
Open the calculator without a preset: 微分

Frequently asked questions

Can I rewrite a quotient as a product?+

Yes. Product and chain rules applied to fg⁻¹ give the same formula.

Should I simplify first?+

Yes when the simplification is valid and excluded domain points remain recorded.

Why is the denominator squared?+

Differentiating g⁻¹ produces g⁻² through the power rule.

Conclusion

Keep f′g−fg′ in order, square the original denominator, and preserve every domain exclusion.

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