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- Chain Rule - CR
- Note
- Quantitative Aptitude
- Placement Preparation
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The Chain Rule The chain rule is the last of the derivative formulas. It is most easily written using variables and the Leibniz notation: CHAIN RULE If C depends on ?, and ? depends on B, then: .C .C .? œ † .B .? .B The idea is that the two .?'s cancel: y .C .? .C † œ y .B .? .B Though this may seem obvious, remember that .B, .C, and .? are “infinitesimals”, which can't really be treated like regular numbers. The Leibniz notation makes the chain rule look obvious, but we are actually on slightly shaky ground when we talk about canceling two infinitesimals. In any case, the chain rule is true, and it is incredibly useful for taking derivatives of complicated formulas. EXAMPLE 1 Find SOLUTION .C if C œ ÈB$ ". .B Let ? œ B$ ". Then C depends on ?, and ? depends on B: C œ È? and ? œ B$ " We take the derivatives of both of these formulas: .C " œ .? #È ? and .? œ $B# .B and then multiply the results together: .C .C .? " œ † œ † $B# .B .? .B #È ? Finally, we can plug in the formula for ? to express .CÎ.B in terms of B: .C " œ † $B# $ È .B # B " è

.C " if C œ ŒB# . .B B & EXAMPLE 2 Find SOLUTION Let ? œ B# " . Then C depends on ?, and ? depends on B: B " C œ ?& and ? œ B# B Therefore: .C " " " œ &?% Œ#B # œ &ŒB# Œ#B # .B B B B í ï .C .? .? .B % è Aside from the power rule, the chain rule is the most important of the derivative rules, and we will be using the chain rule hundreds of times this semester. Because of this, it is important that you get used to the pattern of the chain rule, so that you can apply it in a single step. In any case where: C œ asomethingb& the derivative will always be: .C œ &asomethingb% † athe derivative of the somethingb .B The “something” is what we have been calling ?, but it's fastest not to even think of it as a variable. If you are given a formula like: C œ ˆ)B$ $B #‰ & you should just be able to write down the derivative: .C % œ &ˆ)B$ $B #‰ † ˆ#%B# $‰ .B Similarly, the derivative of: C œ Èsomething is always: .C " œ † athe derivative of the somethingb È .B # something For example, if you are given the formula: C œ È B& B

you should immediately be able to write the derivative: .C " œ † ˆ&B% "‰ & È .B # B B Combining Rules For complicated formulas, you sometimes need to combine different rules together. Here are a few examples: EXAMPLE 3 Find SOLUTION .C & if C œ ˆB# $‰ ÈB. .B Here C is a product of two other quantities: C œ @A, where @ œ ˆB# $‰ and A œ ÈB & We should therefore use the product rule. However, the chain rule will also come into play when we take the derivative of @: .C " & % œ ˆB# $‰ † ÈB † &ˆB# $‰ † #B .B #È B í ðóóóñóóóò ðñò í .A A .@ @ .B .B EXAMPLE 4 Find è .C B$ if C œ . È " B# .B This requires the quotient rule, but the chain rule will come into play when we take the derivative of È" B# : SOLUTION .C œ .B ŠÈ" B# ‹a$B# b aB$ b " B# " † #B È # " B# è

Related Rates The chain rule can also be used to find a relationship between two unknown derivatives. For example, suppose that: C œ ?$ .C ? This is similar to a normal chain rule problem, except that we don't have a formula .B for ? in terms of B. The best we can do is write: What is .C .? œ $?# † .B .B í í .C .? .? .B This follows the same basic pattern as any other chain rule: the derivative of asomethingb$ is $asomethingb# multiplied by the derivative of the something. EXAMPLE 5 Find SOLUTION .C if C œ È?. .B Using the chain rule: .C " .? œ † .B #È? .B í í .C .? .? .B è If the variable B is time, this method can be used to find the relationship between two different rates of change: EXAMPLE 6 The side length P and area E of a square are related by the equation: E œ P# What is the relationship between SOLUTION .E .P and ? .> .> Using the chain rule: .E .P œ #P † .> .> í í .E .P .P .> è

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