From Rogawski ET 2e section 14.6, exercise 1. (a) Calculate the primary derivati

Question

From Rogawski ET 2e section 14.6, exercise 1.

(a) Calculate the primary derivatives

(b) Calculate

(c) Use the Chain Rule to compute

From Rogawski ET 2e section 14.6, exercise 1. Let f(x,y,z)=x^{3}y^{2}+z^{4} and x=s^{3}t^{2}, y=st^{3}, and z=st. Calculate the primary derivatives rac{partial{f}}{partial{x}}= 3x^2y^2 Calculate rac{partial{x}}{partial{s}}= Use the Chain Rule to compute (9s^10t^12)+(2s^10t^11)+(4s^3t^4) In (c) express your answer in terms of the independent variables t,s Help please !!!

Explanation / Answer

c) use chain rule

df/ds = df/dx *dx/ds + df/dy*dy/ds +df/dz *dz/ds

df/dx = 3x^2 y^2 = 3(s^3 t^2)^2 (st^3)^2 = 3 *s^8 t^10

df/dy = 2yx^3 = 2(st^3)(s^3 t^2)^3 = 2s^10 t^9

df/dz = 4z^3 = 4s^3 t^3

dx/ds = 3s^2 t^2

dy/ds = t^3

dz/ds = t

df/ds = 3s^8 t^10 * 3s^2 t^2 + 2s^10 t^9 * t^3 + 4s^3 t^3 * t

= 9 s^10 t^12 + 2 s^10 t^12 + 4 s^3 t^4

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