The chain rule states that if you have two nested functions, where f(x) = g(h(x))f(x)=g(h(x)), the derivative is;
f'(x) = g'(h(x)) * h'(x)
When you have functions with many nested radicals, its often easiest to go through each function step by step.
d/dx sqrt(ln(xe^x))
The first function is the square root function. We can rewrite a square root as;
sqrt(f(x)) = f(x)^(1/2)
Using the power rule;
d/dx f(x)^(1/2) = 1/2 f(x)^(-1/2) *f'(x)
= 1/(2f(x)^(1/2)) *f'(x)
= 1/(2sqrt(f(x))) *f'(x)
Let f(x) = ln(xe^x).
d/dx sqrt(ln(xe^x)) = 1/(2sqrt(ln(xe^x))) d/dx ln(xe^x)
Next we look at the ln. The derivative of ln(f(x)) is 1"/"f(x) * f'(x).
1/(2sqrt(ln(xe^x))) d/dx ln(xe^x) = 1/(2sqrt(ln(xe^x))) 1/(xe^x) d/dx xe^x
The last part of the derivative can be solved using the product rule.
d/dx f(x)g(x) = f'(x)g(x) + f(x)g'(x)
Remember that d/dx e^x = e^x.
1/(2sqrt(ln(xe^x))) 1/(xe^x) d/dx xe^x = 1/(2sqrt(ln(xe^x))) 1/(xe^x) (e^x + xe^x)
We can simplify by canceling out all of the e^x terms
1/(2sqrt(ln(xe^x))) 1/(x color(red)cancel(color(black)(e^x))) (color(red)cancel(color(black)(e^x)) + xcolor(red)cancel(color(black)(e^x))) = (1+x)/(2xsqrt(ln(xe^x)))
