Chapter 2: Q 89 (page 212)

In Exercise $89$ of the Section $2.3$ you used the definition of derivative to prove the quotient rule. Prove it now another way: by writing a quotient $\frac{f}{g}$ as a product and applying the product, power, and chain rules. Point out where you use each rule.

Short Answer

Expert verified

$y = \frac{f}{g} \Rightarrow y g = f \Rightarrow \frac{d}{d x} (y g) = \frac{d}{d x} (f) \Rightarrow y \frac{d g}{d x} + g \frac{d y}{d x} = \frac{d f}{d x} \Rightarrow g \frac{d y}{d x} = \frac{d f}{d x} - y \frac{d g}{d x} \Rightarrow \frac{d y}{d x} = \frac{\frac{d f}{d x} - y \frac{d g}{d x}}{g} \Rightarrow \frac{d}{d x} (\frac{f}{g}) = \frac{\frac{d f}{d x} - \frac{f}{g} \frac{d g}{d x}}{g} \Rightarrow \frac{d}{d x} (\frac{f}{g}) = \frac{g \frac{d f}{d x} - f \frac{d g}{d x}}{g^{2}}$

Hence proved.

Step by step solution

Step 1. Given Information

We have to prove the quotient rule of derivative.

That is we have to prove that :-

$\frac{d}{d x} (\frac{f}{g}) = \frac{g \frac{d f}{d x} - f \frac{d g}{d x}}{g^{2}}$

We have to convert quotient to a product. Then we have to use product, quotient and chain rule to prove this rule.

Step 2. Proof of quotient rule

Consider the following quotient :-

$y = \frac{f}{g} \Rightarrow y g = f$

Take derivative on both sides, then we have :-

$\frac{d}{d x} (y g) = \frac{d}{d x} (f)$

Now use product and chain rule, then we have :-

$y \frac{d g}{d x} + g \frac{d y}{d x} = \frac{d f}{d x} \Rightarrow g \frac{d y}{d x} = \frac{d f}{d x} - y \frac{d g}{d x} \Rightarrow \frac{d y}{d x} = \frac{\frac{d f}{d x} - y \frac{d g}{d x}}{g}$

Put the value of $y = \frac{f}{g}$ , then we have :-

role="math" localid="1648663599119" $\frac{d}{d x} (\frac{f}{g}) = \frac{\frac{d f}{d x} - \frac{f}{g} \frac{d g}{d x}}{g} \Rightarrow \frac{d}{d x} (\frac{f}{g}) = \frac{g \frac{d f}{d x} - f \frac{d g}{d x}}{g^{2}}$

This is the required result.

Hence the quotient rule is proved.

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In Exercise $89$ of the Section $2.3$ you used the definition of derivative to prove the quotient rule. Prove it now another way: by writing a quotient $\frac{f}{g}$ as a product and applying the product, power, and chain rules. Point out where you use each rule.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proof of quotient rule

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In Exercise 89of the Section 2.3you used the definition of derivative to prove the quotient rule. Prove it now another way: by writing a quotient fg as a product and applying the product, power, and chain rules. Point out where you use each rule.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proof of quotient rule

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

In Exercise $89$ of the Section $2.3$ you used the definition of derivative to prove the quotient rule. Prove it now another way: by writing a quotient $\frac{f}{g}$ as a product and applying the product, power, and chain rules. Point out where you use each rule.