# How do you differentiate #y = 1/2 x + 1/4sin2x#?

Use implicit differentiation and the product rule to differentiate.

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To differentiate ( y = \frac{1}{2}x + \frac{1}{4}\sin(2x) ), you can use the rules of differentiation:

- Differentiate each term separately.
- For ( \frac{1}{2}x ), the derivative of ( x ) with respect to ( x ) is ( 1/2 ).
- For ( \frac{1}{4}\sin(2x) ), apply the chain rule. The derivative of ( \sin(2x) ) with respect to ( x ) is ( 2\cos(2x) ).
- Multiply the result of step 3 by the derivative of the inner function, which is ( 2 ).

So, the derivative of ( y ) with respect to ( x ) is:

[ y' = \frac{1}{2} + \frac{1}{4} \times 2 \cos(2x) ]

Simplify:

[ y' = \frac{1}{2} + \frac{1}{2} \cos(2x) ]

Thus, the derivative of ( y ) with respect to ( x ) is ( \frac{1}{2} + \frac{1}{2} \cos(2x) ).

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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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