# How do you differentiate #y = log_2 (x^4sinx)#?

first switch into natural logs as these are calculus friendly

so therefore by the chain rule

by the product rule

so

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To differentiate ( y = \log_2(x^4 \sin(x)) ), use the chain rule and the derivative of the natural logarithm function:

( \frac{dy}{dx} = \frac{1}{\ln(2) \cdot x^4 \sin(x)} \cdot \frac{d}{dx}(x^4 \sin(x)) )

Using the product rule for differentiation, the derivative of ( x^4 \sin(x) ) is:

( \frac{d}{dx}(x^4 \sin(x)) = 4x^3 \sin(x) + x^4 \cos(x) )

So, the final expression for the derivative of ( y ) is:

( \frac{dy}{dx} = \frac{1}{\ln(2) \cdot x^4 \sin(x)} \cdot (4x^3 \sin(x) + x^4 \cos(x)) )

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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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