# Can someone tell me where my error is in finding the derivative of #y=x^(lnx)#?

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I'm using the rule #d/dxa^u = a^u ln a (du)/dx# .

#dy/dx = x^(lnx) * ln x * 1/x#

#=(x^(lnx) ln x)/x#

The correct answer is #=(2x^(lnx) ln x)/x# .

I'm using the rule

#=(x^(lnx) ln x)/x#

The correct answer is

##### 2 Answers

#### Explanation:

I'm pretty fond of logarithmic differentiation. We try to take the derivative of both sides:

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

#lny = lnxlnx#

Now use implicit differentiation and the product rule.

#1/y(dy/dx) = 1/xlnx + 1/xlnx#

#1/y(dy/dx) = 2/xlnx#

#dy/dx = y(2/xlnx)#

#dy/dx= x^lnx(2/xlnx)#

Or

#dy/dx = (2x^lnxlnx)/x#

As required.

Hopefully this helps!

I got

#### Explanation:

You've used the wrong derivative technique!

This can only be used when

But we have an

So we must use a special technique which involves talking the natural log

Given:

**Take #ln# of both sides**

Since

**Differentiate both sides W.R.T #x# (The right side requires the product rule)**

**For the left side:**

**For the right side:**

Product rule:

Let

Thus

Multiply both sides by

**We want to rewrite everything in terms of #x# but we have this #color(red)(y# in the way. However, recall that #color(red)(y)# is defined as #color(red)(y=x^lnx)#**

**Rewriting we get:**