How do you differentiate #f(x)= ln(1x)^4#?
Using the chain rule.
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To differentiate the function ( f(x) = \ln{(1x)^4} ), you can use the chain rule along with the derivative of the natural logarithm function. Here's the process:

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

Identify the inner function: [ u = (1x)^4 ]

Find the derivative of ( u ) with respect to ( x ): [ \frac{du}{dx} = 4(1x)^3 \cdot (1) = 4(1x)^3 ]

Substitute ( u ) and ( \frac{du}{dx} ) into the chain rule formula: [ \frac{d}{dx}[\ln{(1x)^4}] = \frac{1}{(1x)^4} \cdot (4(1x)^3) ]

Simplify the expression: [ \frac{d}{dx}[\ln{(1x)^4}] = \frac{4(1x)^3}{(1x)^4} ]

Further simplify: [ \frac{d}{dx}[\ln{(1x)^4}] = \frac{4}{(1x)} ]
So, the derivative of ( f(x) = \ln{(1x)^4} ) is ( \frac{4}{(1x)} ).
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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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