How do you use logarithmic differentiation to find the derivative of #y=(cosx)^x#?
take natural logs of bothe sides
the #RHS will need the product rule
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To find the derivative of ( y = (\cos(x))^x ) using logarithmic differentiation, follow these steps:

Take the natural logarithm of both sides of the equation: [ \ln(y) = \ln\left((\cos(x))^x\right) ]

Use the properties of logarithms to simplify the expression: [ \ln(y) = x \ln(\cos(x)) ]

Differentiate both sides of the equation with respect to ( x ): [ \frac{d}{dx}(\ln(y)) = \frac{d}{dx}(x \ln(\cos(x))) ]

Apply the chain rule and product rule on the righthand side: [ \frac{1}{y} \frac{dy}{dx} = \ln(\cos(x)) + x \left(\frac{\sin(x)}{\cos(x)}\right) ]

Solve for ( \frac{dy}{dx} ): [ \frac{dy}{dx} = y \left(\ln(\cos(x))  \frac{x\sin(x)}{\cos(x)}\right) ]

Substitute back the original expression for ( y ): [ \frac{dy}{dx} = (\cos(x))^x \left(\ln(\cos(x))  \frac{x\sin(x)}{\cos(x)}\right) ]
So, the derivative of ( y = (\cos(x))^x ) with respect to ( x ) using logarithmic differentiation is: [ \frac{dy}{dx} = (\cos(x))^x \left(\ln(\cos(x))  \frac{x\sin(x)}{\cos(x)}\right) ]
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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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