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.
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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