How do you differentiate f(x)=(ln(x))^x f(x)=(ln(x))x?

2 Answers
Dec 12, 2015

Take the natural log of both sides, then use implicit differentiation ...

Explanation:

Take natural log of both sides, then use the property of logs :

lny=ln(lnx)^x=xln(lnx)lny=ln(lnx)x=xln(lnx)

Now, using implicit differentiation, product and chain rules ...

(1/y)y'=ln(lnx)+(x/lnx)xx1/x=ln(lnx)+1/lnx

Finally, solve for y'

y'=y[ln(lnx)+1/lnx]=(lnx)^x[ln(lnx)+1/lnx]

hope that helped

Dec 12, 2015

(IGNORE)

Explanation:

Rewrite f(x) using the properties of logarithms.

f(x)=xln(x)

Now, to find f'(x), use the product rule.

f'(x)=ln(x)d/dx[x]+xd/dx[ln(x)]

Find each derivative.

d/dx[x]=1

d/dx[ln(x)]=1/x

Plug the derivatives back in.

f'(x)=1(ln(x))+x(1/x)

f'(x)=ln(x)+1