   Chapter 14.5, Problem 4E

Chapter
Section
Textbook Problem

Use the Chain Rule to find dz/dt or dw/dt.4. z = 1 + x y , x = tan t, y = arctan t

To determine

To find: The value of dzdt using Chain Rule if z=1+xy , x=tant and y=arctant .

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 functions z=1+xy . (1)

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

zx=121+xy(y)=y21+xy

Thus, the partial derivative, zx=y21+xy .

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

zx=121+xy(x)=x21+xy

Thus, the partial derivate, zy=x21+xy .

Obtain the partial derivative of x with respect to t .

x=tantdxdt=sec2t=sec2t

Thus, the derivate, dxdt=sec2t

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