Composite exponential function differentiation

Intro

product rule
chain rule
d/dx(ax)=ln(a).ax

Sample Problem

Solution

y=(ax)x
ln(y)=ln(ax)x take (ln) both side
ln(y)=x(ln(ax)) ln(a)m=m.ln(a)
d/dx(ln(y))=d/dx(x.(ln(ax))) differentiate both side
1/y.dy/dx=d/dx(x).ln(ax)+d/dx(ln(ax)).x product rule
=1.ln(ax)+1/ax.ln(a).ax.x chain rule
dy/dx=y.(ln(ax)+x.ln(a)) simplify
dy/dx=(ax)x.(ln(ax)+x.ln(a)) substituting y