How do you differentiate #f(x) = x(x^2 - 2)^(3/2) # using the product rule?

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Answer 1 · 2015-12-12T14:23:29

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

According to the product rule:

#f'(x)=(x^2-2)^(3/2)d/dx[x]+xd/dx[(x^2-2)^(3/2)]#

Find each derivative separately.

#d/dx[x]=1#

#d/dx[(x^2-2)^(3/2)]=3/2(x^2-2)^(1/2)d/dx[x^2-2]#

#=3/2(x^2-2)^(1/2)(2x)=3x(x^2-2)^(1/2)#

The previous rule required the chain rule.

Plug both derivatives back in.

#f'(x)=(x^2-2)^(3/2)+3x^2(x^2-2)^(1/2)#

#f'(x)=(x^2-2)^(1/2)(x^2-2+3x^2)#

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