What is the derivative of (x+1)^x(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)^xy=(x+1)x
Take the natural logarithm of both sides.
ln(y)=ln((x+1)^x)ln(y)=ln((x+1)x)
Rewrite using logarithm rules.
ln(y)=xln(x+1)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))1y(dydx)=ln(x+1)ddx(x)+xddx(ln(x+1))
Note that differentiating
1/y(dy/dx)=ln(x+1)+x(1/(x+1))d/dx(x+1)1y(dydx)=ln(x+1)+x(1x+1)ddx(x+1)
1/y(dy/dx)=ln(x+1)+x/(x+1)1y(dydx)=ln(x+1)+xx+1
Now, solve for
dy/dx=(x+1)^x(ln(x+1)+x/(x+1))dydx=(x+1)x(ln(x+1)+xx+1)