How do you differentiate #f(x)=ln(x(x^2+1)/(2x^3-1)^(1/2)) #?

Answer 1

#d/dx (ln ( (x (x^2+1))/(2x^3-1)^(1/2)) ) = (3x^5-x^3-3x^2+1)/(x(x^2+1)(2x^3-1))#

Note that in general, using the chain rule:

(1) #d/dx ln(f(x)) = (f'(x))/f(x)#

Using the properties of logarithms we have:

#ln ( (x (x^2+1))/(2x^3-1)^(1/2)) = lnx + ln(x^2+1) -1/2ln(2x^3-1)#

and as the derivative is linear:

#d/dx (ln ( (x (x^2+1))/(2x^3-1)^(1/2)) ) = d/dx lnx + d/dx ln(x^2+1) -1/2d/dx ln(2x^3-1)#

Then, based on (1):

#d/dx (ln ( (x (x^2+1))/(2x^3-1)^(1/2)) ) = 1/x + (2x)/(x^2+1) -1/2(6x^2)/(2x^3-1)#
#d/dx (ln ( (x (x^2+1))/(2x^3-1)^(1/2)) ) = 1/x + (2x)/(x^2+1) -(3x^2)/(2x^3-1)#
#d/dx (ln ( (x (x^2+1))/(2x^3-1)^(1/2)) ) = ((2x^3-1)(x^2+1)+2x^2(2x^3-1)-3x^3(x^2+1))/(x(x^2+1)(2x^3-1))#
#d/dx (ln ( (x (x^2+1))/(2x^3-1)^(1/2)) ) = (2x^5-x^2+2x^3+1+4x^5-2x^2-3x^5-3x^3)/(x(x^2+1)(2x^3-1))#
#d/dx (ln ( (x (x^2+1))/(2x^3-1)^(1/2)) ) = (3x^5-x^3-3x^2+1)/(x(x^2+1)(2x^3-1))#
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Answer 2

To differentiate the function ( f(x) = \ln\left(\frac{x(x^2+1)}{(2x^3-1)^{\frac{1}{2}}}\right) ), you would use the chain rule and the properties of logarithmic functions. The steps are as follows:

  1. Apply the quotient rule to differentiate the inside function ( \frac{x(x^2+1)}{(2x^3-1)^{\frac{1}{2}}} ).
  2. Differentiate the numerator and denominator separately.
  3. Apply the chain rule to the denominator when differentiating.
  4. Combine the results using the properties of logarithmic functions.

Here's the result:

[ f'(x) = \frac{1}{\frac{x(x^2+1)}{(2x^3-1)^{\frac{1}{2}}}} \times \left(\frac{d}{dx}\left(\frac{x(x^2+1)}{(2x^3-1)^{\frac{1}{2}}}\right)\right) ]

[ = \frac{1}{\frac{x(x^2+1)}{(2x^3-1)^{\frac{1}{2}}}} \times \left(\frac{(x^2 + 1)(2x^3 - 1)^{\frac{1}{2}} - x \cdot \frac{1}{2}(2x^3 - 1)^{-\frac{1}{2}}(6x^2)}{(2x^3-1)}\right) ]

[ = \frac{(2x^3 - 1)^{\frac{1}{2}}((x^2 + 1) - 3x^2)}{x(x^2+1)} ]

[ = \frac{(2x^3 - 1)^{\frac{1}{2}}(1 - 2x^2)}{x(x^2+1)} ]

[ = \frac{(2x^3 - 1)^{\frac{1}{2}} - 2x^3(2x^3 - 1)^{\frac{1}{2}}}{x(x^2+1)} ]

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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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