Quotient Rule

📖 What is the Quotient Rule?

The quotient rule is used to differentiate a function that is the ratio (division) of two functions. It is sometimes remembered with the mnemonic "low d-high minus high d-low, over low squared."

            d/dx(f/g) = (g · f' − f · g') / g²
          

Where g(x) ≠ 0.

🔤 What Each Symbol Means

f

Numerator function (high)The function on top of the fraction

g

Denominator function (low)The function on the bottom — must not equal zero

f'

Derivative of the numeratord-high

g'

Derivative of the denominatord-low

g²

Denominator squaredlow squared — always positive (when g ≠ 0)

📝 Step-by-Step Example 1

Differentiate: y = sin x / cos x = tan x

1

Identify f and gf = sin x g = cos x

2

Find the derivativesf' = cos x g' = -sin x

3

Apply: (gf' − fg') / g²(cos x · cos x − sin x · (−sin x)) / cos²x

4

Simplify numerator: cos²x + sin²x = 1= 1 / cos²x

✓

Answer: dy/dx = sec²x (confirms tan derivative!)

📝 Step-by-Step Example 2

Differentiate: y = x² / (x + 1)

1

Identify f and gf = x² g = x + 1

2

Find the derivativesf' = 2x g' = 1

3

Apply: (gf' − fg') / g²((x+1)·2x − x²·1) / (x+1)²

4

Expand numerator(2x² + 2x − x²) / (x+1)² = (x² + 2x) / (x+1)²

✓

Answer: dy/dx = x(x + 2) / (x + 1)²

💡 Memory Trick — "Lo D-Hi Minus Hi D-Lo"

"Lo" = g (denominator)
"Hi" = f (numerator)
"D-Hi" = f' "D-Lo" = g'

(Lo · D-Hi − Hi · D-Lo) / Lo²

📖 What is the Quotient Rule?

🔤 What Each Symbol Means

📝 Step-by-Step Example 1

📝 Step-by-Step Example 2

💡 Memory Trick — "Lo D-Hi Minus Hi D-Lo"

🔗 Related Formulas

🎯 Quick Fact