Can someone tell me where my error is in finding the derivative of #y=x^(lnx)#?
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
Since
Differentiate both sides W.R.T
For the left side:
For the right side:
Product rule:
Let
Thus
Multiply both sides by
We want to rewrite everything in terms of
Rewriting we get: