How do you find the derivative of #y=ln(cos(x))# ?

Answer 1
You can find this derivative by applying the Chain Rule, with #cosx# as the inner function, and #lnx# as the outer function.

Process:

To apply the chain rule, we first find the derivative of the outer function, #lnu#, with #u = cosx#. Remember that the derivative of #lnu = 1/u = 1/cosx#.
Now we just need to find the derivative of the inner function, #cosx#, and multiply it by the derivative of the outer function we just found.
Since the derivative of #cosx# is (#-sinx#), we end up with:
#dy/dx = (1/cosx) * (-sinx) = (-sinx/cosx) = -tanx#.
A shorter way to do these is to just know that the derivative of a #ln(u)#-type function is the derivative of the inside over the original of what's inside.
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Answer 2

#dy/dx=-tanx#

#"differentiate using the "color(blue)"chain rule"#
#• d/dx(ln(f(x)))=(f'(x))/(f(x))#
#rArrdy/dx=(-sinx)/(cosx)=-tanx#
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Answer 3

To find the derivative of ( y = \ln(\cos(x)) ), we use the chain rule. The derivative of ( \ln(u) ) with respect to ( u ) is ( \frac{1}{u} ), and the derivative of ( \cos(x) ) with respect to ( x ) is ( -\sin(x) ). So, applying the chain rule, the derivative of ( y ) with respect to ( x ) is:

[ \frac{dy}{dx} = \frac{1}{\cos(x)} \cdot (-\sin(x)) = -\frac{\sin(x)}{\cos(x)} = -\tan(x) ]

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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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