How do you differentiate #f(x)=1/xsqrt(x-1)# using the product rule?

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Answer 1 · 2015-11-24T03:19:21

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

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

According to the product rule:

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

We can figure out both the derivatives here:

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

The next requires a simple use of the chain rule:

#d/dx[(x-1)^(1/2)]=1/2(x-1)^(-1/2)stackrel"= 1"overbrace(d/dx[x-1])=1/(2(x-1)^(1/2))#

Plug them back in:

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

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

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