___ \/ x *log(x)
sqrt(x)*log(x)
Apply the product rule:
f(x)=xf{\left(x \right)} = \sqrt{x}f(x)=x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}dxdf(x):
Apply the power rule: x\sqrt{x}x goes to 12x\frac{1}{2 \sqrt{x}}2x1
g(x)=log(x)g{\left(x \right)} = \log{\left(x \right)}g(x)=log(x); to find ddxg(x)\frac{d}{d x} g{\left(x \right)}dxdg(x):
The derivative of log(x)\log{\left(x \right)}log(x) is 1x\frac{1}{x}x1.
The result is: log(x)2x+1x\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}2xlog(x)+x1
Now simplify:
The answer is:
1 log(x) ----- + ------- ___ ___ \/ x 2*\/ x
-log(x) -------- 3/2 4*x
-2 + 3*log(x) ------------- 5/2 8*x