Question

How do you differentiate `f(x)=x/(1+sqrtx)`?

Answer 1

`f'(x)= (2+sqrtx)/(2(1+sqrtx)^2`

Explanation:

`f(x)" "` is differentiated by applying the quotient rule
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differentiation.
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Quotient Rule:
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`color(blue)((u/v)'=(u'v - v'u)/v^2`
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`f'(x)=(x/(1+sqrtx))'`
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`f'(x)= color(blue)((x'(1+sqrtx)-(1+sqrtx)'x)/(1+sqrtx)^2)`
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`f'(x)= (1(1+sqrtx)-(0+1/(2sqrtx))x)/(1+sqrtx)^2`
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`f'(x)= (1+sqrtx-(1/(2sqrtx))x)/(1+sqrtx)^2`
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`f'(x)= (1+sqrtx-x/(2sqrtx))/(1+sqrtx)^2`
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`f'(x)= ((2sqrtx + 2x-x)/(2sqrtx))/(1+sqrtx)^2`
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`f'(x)= ((2sqrtx+x) /(2sqrtx))/(1+sqrtx)^2`
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`f'(x)= ((sqrtx(2+sqrtx)) /(2sqrtx))/(1+sqrtx)^2`
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`f'(x)= ((2+sqrtx)/2)/(1+sqrtx)^2`
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`f'(x)= (2+sqrtx)/(2(1+sqrtx)^2`