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Assume that x and y are both differentiable functions of t and find the required values of...

Question:

Assume that {eq}x{/eq} and {eq}y{/eq} are both differentiable functions of {eq}t{/eq} and find the required values of {eq}\frac{dy}{dt}{/eq} and {eq}\frac{dx}{dt}{/eq}.

(a) Find {eq}\frac{dy}{dt}{/eq} when {eq}x = 5{/eq}, given that {eq}\frac{dx}{dt} = 3{/eq}.

{eq}\frac{dy}{dt} = {/eq}

(b) Find {eq}\frac{dx}{dt}{/eq} when {eq}x = 8{/eq}, given that {eq}\frac{dy}{dt} = 2 {/eq}.

{eq}\frac{dx}{dt} ={/eq}

MISSING INFORMATION

Derivatives:


For the parametric functions {eq}x=f(t), \ y=g(t) {/eq}, we can calculate the slope of the tangent {eq}\dfrac{dy}{dx} {/eq} as {eq}\boxed{\dfrac{\frac{dy}{dt}}{\frac{dx}{dt}}=\dfrac{dy}{dx}} {/eq}. Derivatives of the functions represent the slope of the tangent to the curves at a given point.

Answer and Explanation: 1

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Let {eq}y=x^2 {/eq}

a) Given:

  • {eq}\dfrac{dx}{dt} = 3 {/eq}
  • {eq}x=5, \dfrac{dy}{dx}_{ (at \ x=5)}=2(5)=10 {/eq}

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Derivatives: The Formal Definition

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Chapter 7 / Lesson 5
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The derivative in calculus is the rate of change of a function. In this lesson, explore this definition in greater depth and learn how to write derivatives.


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