How do you differentiate #y=e^((lnx)^2)#?

Answer 1
Use #d/dx(e^u) = e^u (du)/dx# and the power and chain rule:
#y=e^((lnx)^2)#
#y' = e^((lnx)^2) * d/dx((lnx)^2) = e^((lnx)^2)*2(lnx)(1/x)#
#y' = (2e^((lnx)^2)lnx)/x#

Note on rewriting

Trying to rewrite is helpful in many problems. It is less helpful in this case than one might hope:

#y = e^((lnx)^2)= (e^lnx)^lnx = x^lnx#
And #y=x^lnx# can be differentiated by logarithmic differentiation, but it doesn't seem easier than the approach taken above.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To differentiate ( y = e^{(\ln(x))^2} ), you can use the chain rule. The chain rule states that if ( y = f(g(x)) ), then ( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) ). In this case, let ( f(u) = e^{u^2} ) and ( g(x) = \ln(x) ).

First, find the derivative of ( f(u) = e^{u^2} ), which is ( f'(u) = 2ue^{u^2} ).

Then, find the derivative of ( g(x) = \ln(x) ), which is ( g'(x) = \frac{1}{x} ).

Now, apply the chain rule:

[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = 2(\ln(x))e^{(\ln(x))^2} \cdot \frac{1}{x} = \frac{2\ln(x)}{x}e^{(\ln(x))^2} ]

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7