g(x)=√(x)sinx

Question

Trying to find the derivative using the product rule. Would appreciate if steps are explained.

Answer 1

g(x)=(√x)(sin(x)).

Let u=√x=x^½ and v=sin(x), so du/dx=½x^-½ and dv/dx=cos(x).

dg/dx=d(uv)/dx=vdu/dx+udv/dx (product rule)=(sin(x))(½/√x)+(√x)(cos(x)).

This can be written: dg/dx=sin(x)/(2√x)+cos(x)√x, or:

dg/dx=½sin(x)√x/x+cos(x)√x, or (½sin(x)/x+cos(x))√x.