What is the derivative of #(x+1)^x#?
2 Answers
Explanation:
lets use
now lets differentiate the two sides in respect of
Explanation:
I think the easiest way to do these kinds of problems (with a variable power) without memorizing the formula is through implicit differentiation.
#y=(x+1)^x#
Take the natural logarithm of both sides.
#ln(y)=ln((x+1)^x)#
Rewrite using logarithm rules.
#ln(y)=xln(x+1)#
Differentiate both sides with respect to
The left side will spit out a
#1/y(dy/dx)=ln(x+1)d/dx(x)+xd/dx(ln(x+1))#
Note that differentiating
#1/y(dy/dx)=ln(x+1)+x(1/(x+1))d/dx(x+1)#
#1/y(dy/dx)=ln(x+1)+x/(x+1)#
Now, solve for
#dy/dx=(x+1)^x(ln(x+1)+x/(x+1))#