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What is the derivative of #f(x) = ln(sin^2(x))#?

Answer 1

Scarlett Allison

#d/dxln(sin^2(x)) = 2cot(x)#

We will use the following: The chain rule #d/dxf(g(x)) = f'(g(x)g'(x)#

#d/dxln(x) = 1/x#

#d/dx x^n = nx^(n-1)#

#d/dx sin(x) = cos(x)#

Now, as the function given is a logarithm of a power of the sine function, we will apply the chain rule twice:

#d/dxln(sin^2(x)) = 1/(sin^2(x))*(d/dxsin^2(x))#

#=> d/dxln(sin^2(x)) = 1/(sin^2(x))* 2sin(x)*(d/dx sin(x))#

#=> d/dxln(sin^2(x)) = (2sin(x))/(sin^2(x))cos(x) = (2cos(x))/sin(x)=2cot(x)#

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Answer 2

Aurora Alvarado

To find the derivative of ( f(x) = \ln(\sin^2(x)) ), we'll use the chain rule.

[ \frac{d}{dx} \left( \ln(\sin^2(x)) \right) = \frac{1}{\sin^2(x)} \cdot \frac{d}{dx}(\sin^2(x)) ]

Now, we need to find the derivative of ( \sin^2(x) ):

[ \frac{d}{dx}(\sin^2(x)) = 2\sin(x)\cos(x) ]

So, putting it all together:

[ \frac{d}{dx} \left( \ln(\sin^2(x)) \right) = \frac{1}{\sin^2(x)} \cdot 2\sin(x)\cos(x) = \frac{2\sin(x)\cos(x)}{\sin^2(x)} ]

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Answer from HIX Tutor

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