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

Answer 1

The explanation is given below.

#f(x) =cot(sqrt(x^2-1))#

Let us first write the same as the following.

#y = cot(u)# and #u=sqrt(v)# and #v=x^2-1# The above steps are like breaking the chain into manageable links.

Chain rule would work like this:

#dy/dx = dy/(du) * (du)/(dv) * (dv)/(dx)# Chain rule
#y=cot(u)# #dy/(du) = -sec^2(u)#
#u=sqrt(v)# #(du)/(dv) = 1/(2sqrt(v))#
#v=x^2-1# #(dv)/(dx) = 2x#
# f'(x) = dy/dx = -sec^2(u) * 1/(2sqrt(v)) * 2x# Simplifying
#f'(x) = dy/dx =(- x*sec^2(sqrt(v)))/sqrt(v) # #f'(x) = dy/dx = (-x*sec^2(sqrt(x^2-1))/sqrt(x^2-1))#
#f'(x) = (-x*sec^2(sqrt(x^2-1))/sqrt(x^2-1))# answer
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Answer 2

To differentiate the function (f(x) = \cot(\sqrt{x^2-1})) using the chain rule, follow these steps:

  1. Recognize the composition of functions: The function (f(x) = \cot(\sqrt{x^2-1})) is a composition of several functions:

    • (u(x) = x^2 - 1)
    • (v(u) = \sqrt{u})
    • (w(v) = \cot(v)) So, (f(x) = w(v(u(x)))).
  2. Apply the chain rule: The chain rule states that if a function (y = f(g(x))), then its derivative (y') can be found as (f'(g(x)) \cdot g'(x)). For our composite function, we'll apply the chain rule multiple times.

  3. Find the derivatives of the inner functions:

    • For (u(x) = x^2 - 1), the derivative (u'(x) = 2x).
    • For (v(u) = \sqrt{u}), applying the derivative of a square root function (v'(u) = \frac{1}{2\sqrt{u}}).
    • For (w(v) = \cot(v)), the derivative (w'(v) = -\csc^2(v)).
  4. Apply the derivatives according to the chain rule: [f'(x) = w'(v(u(x))) \cdot v'(u(x)) \cdot u'(x)]

Substitute the derivatives we found: [f'(x) = -\csc^2(\sqrt{x^2-1}) \cdot \frac{1}{2\sqrt{x^2-1}} \cdot 2x]

Simplify: [f'(x) = -\frac{2x}{2\sqrt{x^2-1}} \cdot \csc^2(\sqrt{x^2-1})] [f'(x) = -\frac{x \cdot \csc^2(\sqrt{x^2-1})}{\sqrt{x^2-1}}]

So, the derivative of (f(x) = \cot(\sqrt{x^2-1})) with respect to (x) is (f'(x) = -\frac{x \cdot \csc^2(\sqrt{x^2-1})}{\sqrt{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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