A thin uniform rod of length l cm and a small particle lie on a line separated by a distance of a...

Question:

A thin uniform rod of length l cm and a small particle lie on a line separated by a distance of a cm. If K is a positive constant and F is measured in new tons, the gravitational force between them is

(a) If a is increasing at the rate 2 cm/min when a = 15 and l = 5, how fast is F decreasing?

(b) If l is decreasing at the rate 2 cm/min when a = 15 and l = 5, how fast is F increasing?

Chain Rule For Differentiation:

Let {eq}y {/eq} be a function of {eq}x {/eq} such that {eq}y=\dfrac{f \left( x \right)}{ g \left( x \right)} {/eq}

If {eq}x {/eq} is increasing at a rate of {eq}a \text{cm/min} {/eq}, then we can find the rate of change of {eq}y {/eq} when {eq}x=b {/eq} by differentiating the above relation with respect to {eq}t {/eq} using chain rule as

{eq}\dfrac{{dy}}{{dt}} = \dfrac{{g\left( x \right)\left( {f'\left( x \right)} \right)\dfrac{{dx}}{{dt}} - f\left( x \right)\left( {g'\left( x \right)} \right)\dfrac{{dx}}{{dt}}}}{{{{\left( {g\left( x \right)} \right)}^2}}} {/eq}

Substitute {eq}x=b \text{ and } \dfrac{dx}{dt} = a {/eq} to get {eq}\dfrac{dy}{dt} {/eq}

Answer and Explanation: 1

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Given that a uniform rod of length {eq}l {/eq} cm and a small particle lie on a line separated by a distance of {eq}a {/eq} cm.

For a positive...

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The Chain Rule for Partial Derivatives

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Chapter 14 / Lesson 4
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When evaluating the derivative of composite functions of several variables, the chain rule for partial derivatives is often used. In this lesson, we use examples to explore this method.


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