How do you differentiate #f(x)=e^((x^2+x^(1/2))^(1/2) )# using the chain rule?

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Answer 1 · 2016-10-27T02:38:12

For# x > 0, f'(x)=f(x)(4xsqrtx+1)/(4sqrt(x^2sqrtx+x))#

For f(x) to be real and differentiable, x > 0.

Use #(f(x))'=(e^u)'u'=f(x)u'#,

where u is the exponent function..-

So, #(f(x))'=f(x)(sqrt(x^2+sqrtx))'#

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

#=f(x)(x+1/(4sqrtx))/sqrt(x^2+sqrtx)#

#=f(x)(4xsqrtx+1)/(4sqrt(x^2sqrtx+x))#