How to Use the rule of the appropriate chain and converting w to function of t before differentiating and then Express the answer in s and t?

Question

#w=sqrt(x^2+y^2+z^2), x=cos s t, y = sin s t,z= s^2t#

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Answer 1 · 2018-05-02T18:42:07

# (dw)/(dt) = (s^4 t)/(sqrt( 1 + s^4 t^2)) #

We have:

# w=sqrt(x^2+y^2+z^2)#

Where:

#x=cos st \ \ #, #y = sin st \ \ #, and #z= s^2t#

So converting #w# to a function of #t#, we have:

# w = sqrt( (cos (st))^2 + (sin(st))^2 + (s^2t)^2)#

# \ \ = sqrt( cos^2 (st) + sin^2(st) + s^4 t^2)#

# \ \ = sqrt( 1 + s^4 t^2)#

So then,m differentiating wrt #t#, assuming #s# is a constant, we have:

# (dw)/(dt) = (1/2)( 1 + s^4 t^2)^(-1/2)(2s^4 t) #

# \ \ \ \ \ \ = (s^4 t)/(sqrt( 1 + s^4 t^2)) #