How do you differentiate #y = x * sqrt (4 - x^2)#?

1 Answer
Jan 19, 2016

Answer:

# dy/dx =( 2 (2 - x^2 ))/sqrt(4 - x^2 ) #

Explanation:

differentiate using the 'product rule' and 'chain rule':

#dy/dx = x . d/dx (4 - x^2)^(1/2 )+ (4 - x^2)^(1/2) .d/dx (x) #

#dy/dx = x(1/2 (4 - x^2)^(-1/2) .d/dx (4 - x^2 ) )+ (4 - x^2 )^(1/2) . 1 #

#dy/dx = x ( 1/2 (4 - x^2 )^(-1/2) .(-2x)) + (4 -x^2)^(1/2)#

#dy/dx = - x^2 (4 - x^2)^(-1/2) + (4 - x^2 )^(1/2) #

[ common factor of # (4 - x^2 )^(-1/2) ] #

#dy/dx = (4 - x^2 )^(-1/2) [ -x^2 + 4 - x^2] #

#dy/dx = (4 - x^2 )^(-1/2) . (4 - 2x^2) #

#rArr dy/dx =( 2(2 - x^2))/sqrt(4 - x^2)#