What is the second derivative of #f(x)= ln sqrt(x+e^x)#?

Answer 1

#f''(x) = (2xe^x - 3e^x - 2)/(2x + e^x)^2#

If we rewrite using laws of logarithms, we get:

#f(x) = ln(x + e^x)^(1/2)= 1/2ln(x + e^x)#

By the chain rule, we get:

#f'(x) = 1/(2(x + e^x)) * (e^x + 1) = (e^x + 1)/(2x + e^x)#

Now by the quotient rule, we get:

#f''(x) = (e^x(2x + e^x) - (2 + e^x)(e^x + 1))/(2x + e^x)^2#
#f''(x) = (2xe^x + e^(2x) - (2e^x + e^(2x) + e^x + 2))/(2x + e^x)^2#
#f''(x) = (2xe^x + e^(2x) - 2e^x - e^(2x) - e^x - 2)/(2x + e^x)^2#
#f''(x) = (2xe^x - 3e^x - 2)/(2x + e^x)^2#

Hopefully this helps!

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 second derivative of ( f(x) = \ln\sqrt{x + e^x} ), you first need to find the first derivative and then differentiate it again.

Given: [ f(x) = \ln\sqrt{x + e^x} ]

First, let's find the first derivative: [ f'(x) = \frac{d}{dx}\left(\ln\sqrt{x + e^x}\right) ]

Using the chain rule and the derivative of natural logarithm: [ f'(x) = \frac{1}{\sqrt{x + e^x}} \cdot \frac{1}{2\sqrt{x + e^x}} \cdot \frac{d}{dx}(x + e^x) ]

[ = \frac{1}{2(x + e^x)} \cdot \frac{1}{\sqrt{x + e^x}} \cdot (1 + e^x) ]

Now, we'll find the second derivative: [ f''(x) = \frac{d}{dx}\left(\frac{1}{2(x + e^x)} \cdot \frac{1}{\sqrt{x + e^x}} \cdot (1 + e^x)\right) ]

[ = \frac{d}{dx}\left(\frac{1 + e^x}{2(x + e^x) \sqrt{x + e^x}}\right) ]

Using the quotient rule: [ = \frac{(2(x + e^x) \sqrt{x + e^x})\cdot 0 - (1 + e^x)\cdot \left(\frac{2\sqrt{x + e^x} + (1 + e^x)\frac{1}{2\sqrt{x + e^x}}}{(2(x + e^x) \sqrt{x + e^x})^2}\right)}{(2(x + e^x) \sqrt{x + e^x})^2} ]

[ = -\frac{(1 + e^x)(2\sqrt{x + e^x} + (1 + e^x)\frac{1}{2\sqrt{x + e^x}})}{4(x + e^x)(x + e^x)} ]

[ = -\frac{(1 + e^x)\left(2\sqrt{x + e^x} + \frac{1 + e^x}{2\sqrt{x + e^x}}\right)}{4(x + e^x)(x + e^x)} ]

[ = -\frac{(1 + e^x)\left(4\sqrt{x + e^x} + 1 + e^x\right)}{8(x + e^x)(x + e^x)} ]

Thus, the second derivative of ( f(x) = \ln\sqrt{x + e^x} ) is: [ f''(x) = -\frac{(1 + e^x)\left(4\sqrt{x + e^x} + 1 + e^x\right)}{8(x + e^x)(x + e^x)} ]

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