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Answer 1 · 2018-01-11T07:08:59

#=> (dy)/(dx) = -8x^(-2x) ( ln(x)+1) #

# y = x^(-2x) #

Taking natural logarithms...

#lny = ln x^(-2x) #

Using the law of:

#color(red)(log_gamma alpha^beta -= beta log_gamma alpha #

#=> lny = -2xln(x) #

Now applying implicit differnetiation....

#=> 1/y * (dy)/(dx) = -2ln(x) -2x*(1/x) #

#=> 1/y * (dy)/(dx) = -2ln(x) - 2 #

#=> (dy)/(dx) = -2y ( ln(x) + 1 ) #

#=> d/(dx) ( x^(-2x) )= -2x^(-2x) ( ln(x)+1) #

#color(blue)(=> d/(dx) ( 4x^(-2x) )= -8x^(-2x) ( ln(x)+1) #

#y = 4x^(-2x) #:

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#(dy)/(dx) = -8x^(-2x) ( ln(x) +1): #

enter image source here