   Chapter 14.5, Problem 1E

Chapter
Section
Textbook Problem

Use the Chain Rule to find dz/dt or dw/dt.1. z = xy3 – x2y, x = t2 + 1, y = t2 − 1

To determine

To find: The value of dzdt by using Chain Rule if z=xy3x2y , x=t2+1 and y=t21 .

Explanation

Chain Rule:

“Suppose that z=f(x,y) is a differentiable function of x and y , where x=g(t)andy=h(t) are both differentiable functions of t . Then, z is differentiable function of t and dzdt=zxdxdt+zydydt ”.

Calculation:

The given function is, z=xy3x2y (1)

Take partial derivative with respect to x of the equation (1),

zx=x(xy3x2y)=x(xy3)x(x2y)=(1)y3(2x)y=y32xy

Thus, the partial derivate, zx=y32xy .

Take partial derivative with respect to y of the equation (1),

zy=y(xy3x2y)=y(xy3)x(x2y)=x(3y2)x2(1)=3xy2x2

Thus, the partial derivate, zy=3xy2x2

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