Find dy/dx by implicit differentiation for #sqrt(5x + y)= 3 + x^2y^2# ?

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Answer 1 · 2018-07-17T21:02:37

#y'=(4xy^2sqrt(5x+y)-5)/(1-x^2ysqrt(5x+y))#

Using the chain rule by assuming #y=y(x)# we get

#1/2(5x+y)^(-1/2)(5+y')=2xy^2+2x^2yy'#

multiplying both sides by #2sqrt(5x+y)# we obtain

#5+y'=4xy^2sqrt(5x+y)+4x^2yy'sqrt(5x+y)#
solving for #y'#

#y'(1-4x^2ysqrt(5x+y))=4x^2ysqrt(5x+y)-5#

so

#y'=(4xy^2sqrt(5x+y)-5)/(1-4x^2ysqrt(5x+y))#

if #1-4x^2ysqrt(5x+y)ne 0#