find the der of
y= x^(lnx)
thanks
Use the function of a function rule, namely
dy/dx = dy/dy . du/dx
substitute u=log(x)
and find the derivative of
d(x^u)/du then multiply by du/dx.
ummm.. i don't understand your language!
can you explain that in baby terms? thanks!
how about this
take ln of both sides
ln y = ln(x^lnx)
ln y = lnx(lnx) = (lnx)^2
now differentiate implicitly
(dy/dx)/y = 2lnx * 1/x
dy/dx = y(2lnx)/x
= (x^lnx) (2lnx)/x
somebody please check this, I have been making some silly mistakes today.
did u get
e^(lnx) / x^2 ??
I will try with an example.
Use the function of a function rule, namely
dy/dx = dy/dy . du/dx
substitute u=log(x)
and find the derivative of
d(x^u)/du then multiply by du/dx.
Suppose y=sin(x²)
We know how to differentiate sin(x), and how to do x², but we're not sure how to find the derivative of sin(x²).
So let u=x²
then
y=sin(u)
dy/du = cos(u)
We also know that
du/dx = d(x²)/dx = 2x
So, by the chain rule,
dy/dx = dy/du.du/dx = 2xcos(x²)
You can apply the same procedure to your problem using u=ln(x).
For Carolina,
There is no e^x in the answer.
The derivative of log(x) is 1/x.
Also,
Reiny's answer is right.
Mine comes up to
2*x^(log(x)-1)*log(x)
which is the same thing.
To find the derivative of the given function, we can use logarithmic differentiation.
Step 1: Take the natural logarithm of both sides of the equation to simplify the equation:
ln(y) = ln(x^(lnx))
Step 2: Use the properties of logarithms to simplify further:
ln(y) = (lnx) * ln(x)
Step 3: Differentiate both sides of the equation implicitly with respect to x:
d/dx[ln(y)] = d/dx[(lnx) * ln(x)]
Step 4: Apply the chain rule on the left side of the equation:
1/y * dy/dx = (1/x) * ln(x) + (lnx) * d/dx[ln(x)]
Step 5: Rewrite the equation with respect to dy/dx:
dy/dx = y * (1/x) * ln(x) + y * (lnx) * d/dx[ln(x)]
Step 6: We can find the derivative d/dx[ln(x)] by differentiating ln(x):
d/dx[ln(x)] = 1/x
Step 7: Substitute the value back into the equation:
dy/dx = y * (1/x) * ln(x) + y * (lnx) * (1/x)
Step 8: Replace y with its original value, x^(lnx):
dy/dx = x^(lnx) * (1/x) * ln(x) + x^(lnx) * (lnx) * (1/x)
Step 9: Simplify the expression:
dy/dx = x^(lnx - 1) * ln(x) + x^(lnx) * (lnx/x)
So, the derivative of y = x^(lnx) is dy/dx = x^(lnx - 1) * ln(x) + x^(lnx) * (lnx/x).