What is the second derivative of #f(x)= ln (x^3+e^x)#?

Answer 1

#f''(x)=((x^3+e^x)(6x+e^x)-(3x^2+e^x)^2)/(x^3+e^x)^2#

We first use the log rule to find

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

We now use the quotient rule to find

#f''(x)=((x^3+e^x)(6x+e^x)-(3x^2+e^x)(3x^2+e^x))/(x^3+e^x)^2#
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Answer 2

To find the second derivative of ( f(x) = \ln(x^3 + e^x) ), follow these steps:

  1. Find the first derivative of ( f(x) ) using the chain rule and the derivative of the natural logarithm function.
  2. Once you have the first derivative, differentiate it again to find the second derivative.

Here's the process:

  1. First derivative: [ f'(x) = \frac{d}{dx}(\ln(x^3 + e^x)) ] [ f'(x) = \frac{1}{x^3 + e^x} \cdot \frac{d}{dx}(x^3 + e^x) ] [ f'(x) = \frac{1}{x^3 + e^x} \cdot (3x^2 + e^x) ]

  2. Second derivative: [ f''(x) = \frac{d}{dx}\left(\frac{1}{x^3 + e^x} \cdot (3x^2 + e^x)\right) ] [ f''(x) = \frac{d}{dx}\left(\frac{3x^2 + e^x}{x^3 + e^x}\right) ] [ f''(x) = \frac{(6x + e^x)(x^3 + e^x) - (3x^2 + e^x)(3x^2)}{(x^3 + e^x)^2} ] [ f''(x) = \frac{(6x + e^x)(x^3 + e^x) - 9x^4 - 3x^2e^x}{(x^3 + e^x)^2} ]

So, the second derivative of ( f(x) = \ln(x^3 + e^x) ) is given by: [ f''(x) = \frac{(6x + e^x)(x^3 + e^x) - 9x^4 - 3x^2e^x}{(x^3 + e^x)^2} ]

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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.

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