   Chapter 14, Problem 35RE

Chapter
Section
Textbook Problem

If u = x2y3 + z4, where x = p + 3p2, y = pep, and z = p sin p, use the Chain Rule to find du/dp.

To determine

To find: The value of dudp using Chain Rule if x=p+3p2 , y=pep and z=psinp .

Explanation

Given:

The function u=x2y3+z4 .

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 function is u=x2y3+z4 (1)

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

ux=x(x2y3+z4)=(2x)y3+0=2xy3

Thus, the partial derivate, ux=2xy3 .

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

uy=y(x2y3+z4)=x2(3y2)+0=3x2y2

Thus, the partial derivate, uy=3x2y2 .

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

uz=z(x2y3+z4)=0+4z3=4z3

Thus, the partial derivate, uz=4z3

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