Question: how do I find the derivative of

x* ln(x+(e^2))^2

x^lnx

x^(e^(-x^2))

Hi Claudia,

I'll do the second problem for you to give you an idea how to approach the third problem.

Let

z = x^ln(x)

Take the natural log of both sides and use the property that ln(a^b) = b ln(a).

ln(z) = ln(x^ln(x)) = ln(x) ln(x) = [ln(x)]²

Now differentiate to get

  ^z'/_z = 2 [ln(x)]  ¹/_x 

Thus

z' = z 2 [ln(x)]  ¹/_x  = 2 x^{ln(x)  ln(x)}/_x

Apply the same technique to the third problem. For the first problem the first step is to use the property of the log that I used above.

Penny