How do you find the first and second derivative of #ln(x^(1/2))#?

Answer 1

For the first derivative you can use the chain rule:

#(dln(x^(1/2)))/dx= (dln(x^(1/2)))/(d(x^(1/2)))*(dx^(1/2))/dx = 1/x^(1/2)*1/2x^(-1/2)=1/(2x)#

but you can also observe that:

#ln(x^(1/2)) = 1/2lnx#

Either way the second derivative is:

#(d^((2))ln(x^(1/2)))/(d^2x)=(d(1/(2x)))/dx=-1/(2x^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 2

To find the first derivative of ln(x^(1/2)), you apply the chain rule, which states that the derivative of ln(u) is (1/u) * u', where u' is the derivative of u with respect to x. Here, u = x^(1/2). Thus, the first derivative of ln(x^(1/2)) is (1/(x^(1/2))) * (1/2) * x^(-1/2), which simplifies to (1/(2x)).

To find the second derivative, you differentiate the first derivative with respect to x. The derivative of (1/(2x)) is -1/(2x^2). So, the second derivative of ln(x^(1/2)) is -1/(2x^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