How do you use the chain rule to differentiate #f(x)=ln(cosx)#?
 tan x
Derivative ln (cos x) Note Derivative ln x = 1 / x So Derivative ln (cos x) = 1 / cos x But we must also take the derivative of the argument [cos x] which is  sin x Thus Derivative Ln (cos x) =  sin x / cos x =  tan x
You can check this by Integrating tan x with U Substitution to obtain Ln (cos x)
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To differentiate ( f(x) = \ln(\cos(x)) ) using the chain rule:

Identify the outer function and the inner function.
 The outer function is ( \ln(x) ).
 The inner function is ( \cos(x) ).

Differentiate the outer function with respect to the inner function.
 The derivative of ( \ln(x) ) is ( \frac{1}{x} ).

Differentiate the inner function with respect to ( x ).
 The derivative of ( \cos(x) ) is ( \sin(x) ).

Multiply the results of steps 2 and 3.
 ( \frac{1}{\cos(x)} ) * ( \sin(x) ) = ( \frac{\sin(x)}{\cos(x)} ).
Thus, the derivative of ( f(x) = \ln(\cos(x)) ) is ( f'(x) = \frac{\sin(x)}{\cos(x)} ), which can be simplified to ( f'(x) = \tan(x) ).
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To differentiate ( f(x) = \ln(\cos(x)) ) using the chain rule, follow these steps:

Identify the outer function and the inner function: The outer function is ( \ln(u) ), where ( u = \cos(x) ) is the inner function.

Compute the derivative of the outer function: The derivative of ( \ln(u) ) with respect to u is ( \frac{1}{u} ).

Compute the derivative of the inner function: The derivative of ( \cos(x) ) with respect to x is ( \sin(x) ).

Apply the chain rule formula: The chain rule states that ( \frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} ).

Substitute the derivatives into the chain rule formula: ( \frac{d}{dx}[\ln(\cos(x))] = \frac{1}{\cos(x)} \cdot (\sin(x)) ).

Simplify the expression: ( \frac{d}{dx}[\ln(\cos(x))] = \frac{\sin(x)}{\cos(x)} ).

Simplify further using trigonometric identities: ( \frac{d}{dx}[\ln(\cos(x))] = \tan(x) ).
Therefore, the derivative of ( f(x) = \ln(\cos(x)) ) with respect to x is ( \tan(x) ).
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To differentiate ( f(x) = \ln(\cos x) ) using the chain rule, follow these steps:
 Identify the outer function and the inner function. In this case, the outer function is the natural logarithm (( \ln )), and the inner function is ( \cos x ).
 Differentiate the outer function with respect to the inner function. The derivative of ( \ln(u) ) with respect to ( u ) is ( \frac{1}{u} ).
 Differentiate the inner function with respect to ( x ). The derivative of ( \cos x ) with respect to ( x ) is ( \sin x ).
 Combine the results using the chain rule formula: ( \frac{d}{dx}[\ln(\cos x)] = \frac{1}{\cos x} \cdot (\sin x) ).
 Simplify the expression if necessary.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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