How do you differentiate #f(x)=sin(1/sqrt(3x^2-4) ) # using the chain rule?

Answer 1

#f'(x) = - cos( 1 / (3x^2 - 4)) * (6x) / (2 sqrt((3x^2-4)^3))#

Let's break down your function using the chain rule:

#f(x) = sin(color(blue)(1/sqrt(3x^2-4)) ) = sin color(blue)(u)#
where #" "u = 1 / sqrt(color(orange)(3x^2-4)) = 1 / sqrt(color(orange)(v)) #
where #" " v = 3x^2 - 4#

According to the chain rule, the derivative is:

#f'(x) = [sin u]' * u' = [sin u]' * [1 / sqrt(v)]' * v' #

Let's compute the derivatives of those terms!

# [sin u]' = cos u = cos( 1 / (3x^2 - 4))#
#u' = [1 / sqrt(v)]' = [v^(-1/2)]' = -1/2 v^(-3/2) = - 1 / (2 sqrt(v^3)) = - 1 / (2 sqrt((3x^2-4)^3))#
#v' = [3x^2 - 4]' = 6x#

Thus, your derivative is:

#f'(x) = [sin u]' * [1 / sqrt(v)]' * v' #
# = cos( 1 / (3x^2 - 4)) * (- 1 / (2 sqrt((3x^2-4)^3))) * 6x#
# = - cos( 1 / (3x^2 - 4)) * (6x) / (2 sqrt((3x^2-4)^3))#
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Answer 2

To differentiate ( f(x) = \sin\left(\frac{1}{\sqrt{3x^2 - 4}}\right) ) using the chain rule, follow these steps:

  1. Identify the outer function and the inner function. In this case, the outer function is (\sin(x)) and the inner function is (\frac{1}{\sqrt{3x^2 - 4}}).

  2. Compute the derivative of the outer function with respect to its input. The derivative of (\sin(x)) is (\cos(x)).

  3. Compute the derivative of the inner function with respect to (x). Let (u = 3x^2 - 4). Then (\frac{du}{dx} = 6x). Now, apply the chain rule to find (\frac{d}{dx}\left(\frac{1}{\sqrt{3x^2 - 4}}\right)).

    [ \frac{d}{dx}\left(\frac{1}{\sqrt{3x^2 - 4}}\right) = \frac{d}{du}\left(\frac{1}{\sqrt{u}}\right) \times \frac{du}{dx} ]

    Simplify (\frac{d}{du}\left(\frac{1}{\sqrt{u}}\right)) to get (-\frac{1}{2u^{3/2}}). So, the derivative of the inner function is (-\frac{1}{2(3x^2 - 4)^{3/2}} \times 6x).

  4. Multiply the derivatives of the outer and inner functions.

  5. The result is the derivative of (f(x)) with respect to (x).

The final derivative is:

[ f'(x) = \cos\left(\frac{1}{\sqrt{3x^2 - 4}}\right) \times \left(-\frac{1}{2(3x^2 - 4)^{3/2}} \times 6x\right) ]

Simplify if necessary.

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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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