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A) Consider the function z = 4x^3 - 2*cos y^2 where x = e^t, y = t^2 - 5. Compute...

Question:

A) Consider the function {eq}z = 4x^3 - 2 \cos y^2 {/eq} where {eq}x = e^t, \; y = t^2 - 5 {/eq}. Compute {eq}\frac{\partial z}{\partial t} {/eq}.

B) Find {eq}\frac{\mathrm{d} y}{\mathrm{d} x} {/eq} if {eq}\ln x^2 + y^2 = 3 {/eq}.

Chain Rule of Partial Derivatives:

Given a function f(x,y) and x= g(t), y=h(t), then {eq}\frac{\partial f}{\partial t} {/eq} is given by {eq}\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}\\ {/eq}Use this to get the partial derivative of given function.

Answer and Explanation: 1

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A) {eq}z = 4x^3-2\cos (y^2) {/eq} and {eq}x =e^t,\ y = t^2-5 {/eq}

{eq}\displaystyle \frac{\partial z}{\partial t}=\frac{\partial z}{\partial...

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The Chain Rule for Partial Derivatives

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Chapter 14 / Lesson 4
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This lesson defines the chain rule. It goes on to explore the chain rule with partial derivatives and integrals of partial derivatives.


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