Consider the function y = x^x √x 1.Express the above function as logy = (x + 1/2)logx 2. Find dy/dx

Question

Consider the function y = x^x \(\sqrt{x}\)

1. Express the above function as logy = \((x+\frac{1}{2})\) logx
2. Find \(\frac{dy}{dx}\)

Answer 1

1. Given, 

y = x^{x \(\sqrt{x}\)}. Take log on both sides,

logy = log(x^x.\(\sqrt{x}\)) = log x^x + log\(\sqrt{x}\) = x logx + \(\frac{1}{2}\)logx = (x + \(\frac{1}{2}\)) logx

2. We have, logy = (x + \(\frac{1}{2}\)) logx Differentiating w.r.t x, we get,