How do you find the derivative of #-ln(x-(x^2+1)^(1/2))#?

Answer 1

The derivative of #-ln(x - (x^2+1)^(1/2))# with respect to x is #1/sqrt(x^2 +1).# To solve this use the chain rule carefully. See the explanation for details.

Starting with:

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

use the chain rule to get:

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

use the chain rule once again on the remaining derivative:

#= -1/(x-(x^2+1)^(1/2)) * (1 - 1/2(x^2+1)^(-1/2)(2x))#

Simplify:

#= (x/(x^2+1)^(1/2) -1 )/(x-(x^2+1)^(1/2))#
Note that #1=(x^2+1)^(1/2) / (x^2+1)^(1/2)#, then substitute this for 1:
#= (x/(x^2+1)^(1/2) - (x^2+1)^(1/2) / (x^2+1)^(1/2) )/(x-(x^2+1)^(1/2))#
#= ((x - (x^2+1)^(1/2)) / (x^2+1)^(1/2) )/(x-(x^2+1)^(1/2))#
#= ((x - (x^2+1)^(1/2)) / (x^2+1)^(1/2) ) divide (x-(x^2+1)^(1/2))#
#= ((x - (x^2+1)^(1/2)) / (x^2+1)^(1/2) ) * 1/(x-(x^2+1)^(1/2))#
#= (x - (x^2+1)^(1/2)) / (x-(x^2+1)^(1/2) ) * 1/((x^2+1)^(1/2))#
#= 1 * 1/((x^2+1)^(1/2)) #
#= 1/((x^2+1)^(1/2)) = 1/sqrt((x^2+1)).#

Finally:

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

If you have any questions about the use of the chain rule or any other part of this solution, then please ask.

Rory.

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Answer 2

To find the derivative of (-\ln(x-(x^2+1)^{1/2})), you can apply the chain rule and the derivative of the natural logarithm function. The derivative is:

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

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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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