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Chain Rule

Rumman Ansari July 28, 2024 506 views

Chain Rule Formulas

Chain Rule Basics

The chain rule is used to solve problems involving multiple quantities that are related to each other.

Direct Proportion

If two quantities \( A \) and \( B \) are directly proportional, then:

\[ \frac{A_1}{B_1} = \frac{A_2}{B_2} \]

Indirect (Inverse) Proportion

If two quantities \( A \) and \( B \) are inversely proportional, then:

\[ A_1 \times B_1 = A_2 \times B_2 \]

Combined Proportion

For combined proportion, where a quantity \( Q \) depends on multiple directly or inversely proportional quantities:

\[ Q \propto \frac{A \times B}{C \times D} \]

where \( A \) and \( B \) are directly proportional to \( Q \), and \( C \) and \( D \) are inversely proportional to \( Q \).

Problem-Solving Using Chain Rule

Steps to solve problems using the chain rule:

Identify the quantities and their relationships (direct or inverse).
Set up the proportion equations based on the relationships.
Solve the equations to find the required quantity.

Example Problem

Suppose \( A \) varies directly as \( B \) and inversely as \( C \). If \( A_1 = 6 \) when \( B_1 = 4 \) and \( C_1 = 2 \), find \( A_2 \) when \( B_2 = 8 \) and \( C_2 = 3 \).

\[ \frac{A_1}{A_2} = \frac{B_1 \times C_2}{B_2 \times C_1} \] \[ \frac{6}{A_2} = \frac{4 \times 3}{8 \times 2} \] \[ \frac{6}{A_2} = \frac{12}{16} = \frac{3}{4} \] \[ A_2 = 6 \times \frac{4}{3} = 8 \]

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