How do you differentiate #f(x)=x^2*sqrt(x-2)# using the product rule?

1 Answer
Jim G.
Jan 19, 2016

# f'(x) = (5x^2 - 8x)/(2sqrt(x - 2)) #

Explanation:

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

rewrite f(x) = # x^2sqrt(x - 2) = x^2.(x - 2 )^(1/2) #

f'(x) = #x^2 d/dx(x - 2 )^(1/2) + (x - 2 )^(1/2) d/dx (x^2) #

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

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

# = 1/2 x^2 (x - 2 )^(-1/2) + 2x (x - 2 )^(1/2) #

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

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

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

<# = (x - 2 )^(-1/2) [ 1/2 (x^2 + 4x^2 - 8x )] #

#rArr f'(x) = (x - 2 )^(-1/2) 1/2 ( 5x^2 - 8x ) = (5x^2 - 8x )/(2sqrt(x - 2 ))#

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