Back to Basics (Mathematics!) : If an expression contains square root or fraction , how will you decide whether to apply Product Rule or Chain Rule ?

When an expression contains square roots or fractions, the choice between the chain rule and the product rule still depends on whether the functions are nested or multiplied.

To make differentiation easier, always rewrite square roots as fractional exponents (√x = x1/2) and fractions using negative exponents (1/x = x-1) before applying either rule.

Here is how you handle square roots and fractions with both rules.

1. Identify Rules for Square Roots

Chain Rule (Nested Square Root)

Use the chain rule when an entire multi-term expression sits inside the square root.

Example: y = √(5x³ + 2)

Rewrite: y = (5x³ + 2)^1/2

Step 1: Differentiate Outside Function

Bring down the exponent 1/2 and subtract 1 from the power. Leave the inside unchanged.

(1/2)(5x³ + 2)^-1/2

Step 2: Multiply by Inside Derivative

The derivative of the inside (5x³ + 2) is 15x². Multiply this to the outside derivative.

dy/dx = (1/2)(5x³ + 2)^-1/2 · (15x²)

Step 3: Simplify and Rewrite

dy/dx = (15x²)/(2√(5x³ + 2))

Product Rule (Multiplied Square Root)

Use the product rule when a square root is an independent term multiplying another distinct function of x.

Example: y = √x · ln(x)

Rewrite: y = x^1/2 · ln(x)

Step 1: Set up Parts

First function (f): x^1/2 ⇒ f' = (1/2)x^-1/2 = 1/(2√x)

Second function (g): ln(x) ⇒ g' = 1/x

Step 2: Apply Product Formula

Multiply f' · g + f · g':

dy/dx = (1/(2√x))(ln(x)) + (√x)(1/x)

Step 3: Simplify and Rewrite

dy/dx = ln(x)/(2√x) + √x/x = (ln(x) + 2)/(2√x)

2. Identify Rules for Fractions

Chain Rule (Nested Fraction)

Use the chain rule when a fraction is nested inside another power or function, or when the entire denominator can be raised to a negative exponent.

Example: y = 1/(x² + 4)

Rewrite: y = (x² + 4)^-1

Step 1: Differentiate Outside Function

Bring down -1 and decrease the power to -2.

-1(x² + 4)^-2

Step 2: Multiply by Inside Derivative

The derivative of the inside (x² + 4) is 2x.

dy/dx = -1(x² + 4)^-2 · (2x)

Step 3: Simplify and Rewrite

dy/dx = -2x/(x² + 4)²

Product Rule (Multiplied Fraction)

Use the product rule instead of the quotient rule when you rewrite a fractional term as a negative power multiplying another function.

Example: y = e^x/x³

Rewrite: y = e^x · x^-3

Step 1: Set up Parts

First function (f): e^x ⇒ f' = e^x

Second function (g): x^-3 ⇒ g' = -3x^-4

Step 2: Apply Product Formula

Multiply f' · g + f · g':

dy/dx = (e^x)(x^-3) + (e^x)(-3x^-4)

Step 3: Simplify and Rewrite

dy/dx = e^x/x³ − 3e^x/x⁴ = e^x(x − 3)/x⁴

Side-by-Side Structural Summary

Structure Type	Function Appearance	Rule Choice	Rewrite Strategy
Nested Root	y = √expression	Chain Rule	(expression)1/2
Multiplied Root	y = √x · f(x)	Product Rule	x1/2 · f(x)
Nested Fraction	y = 1/expression	Chain Rule	(expression)-1
Multiplied Fraction	y = f(x) · 1/g(x)	Product Rule	f(x) · (g(x))-1