How do you differentiate #f(x) = ln((1-x^2)^(-1/2) )?# using the chain rule?

Answer 1

#f'(x)=x(1-x^2)^(-1)#

If #f(x)=ln(g(x))# Then #f'(x)=(g'(x))/(g(x))#
#g(x)=(1-x^2)^(-1/2)=(h(x))^n# #g'(x)=n* h'(x)* h(x)^(n-1)# #h'(x)=-2x# #g'(x)=-1/2*-2x*(1-x^2)^(-3/2)=x(1-x^2)^(-3/2)#
#f'(x)=(x(1-x^2)^(-3/2))/((1-x^2)^(-1/2))=(x(1-x^2)^(1/2))/((1-x^2)^(3/2))=x/((1-x^2)^(2/2))=x/((1-x^2))=x(1-x^2)^(-1)#
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Answer 2

To differentiate f(x) = ln((1-x^2)^(-1/2)) using the chain rule, follow these steps:

  1. Identify the outer function and the inner function.
  2. Differentiate the outer function with respect to the inner function.
  3. Multiply by the derivative of the inner function with respect to x.
  4. Combine the results to find the derivative of the composite function.

Let u = (1-x^2)^(-1/2). Then, f(x) = ln(u).

Now, differentiate ln(u) with respect to u: d(ln(u))/du = 1/u.

Next, differentiate u with respect to x: du/dx = d((1-x^2)^(-1/2))/dx.

Use the chain rule to find du/dx: du/dx = (-1/2)(1-x^2)^(-3/2)(-2x) = x/(1-x^2)^(3/2).

Multiply the derivatives: d(ln(u))/dx = (1/u) * (x/(1-x^2)^(3/2)).

Substitute u back in: d(ln((1-x^2)^(-1/2)))/dx = (1/(1-x^2)^(1/2)) * (x/(1-x^2)^(3/2)).

So, the derivative of f(x) = ln((1-x^2)^(-1/2)) with respect to x using the chain rule is: d(ln((1-x^2)^(-1/2)))/dx = x/(1-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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