# How do you differentiate #y=2^(sinpix)#?

Now we differentiate :

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use lograthmic differentiation

differentiate implicitly

tidying up

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To differentiate ( y = 2^{\sin(\pi x)} ), use the chain rule:

- Find the derivative of the outer function with respect to its inner function.
- Multiply it by the derivative of the inner function.

The derivative of (2^u) with respect to (u) is (2^u \cdot \ln(2)).

Applying this rule:

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

Now, differentiate the inner function (\sin(\pi x)) with respect to (x):

( \frac{d}{dx}(\sin(\pi x)) = \pi \cos(\pi x) )

So, putting it all together:

( \frac{dy}{dx} = 2^{\sin(\pi x)} \cdot \ln(2) \cdot \pi \cos(\pi 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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