You pick one apple from the basket.
2 answers:
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The derivative is given by the limit
You have
Use the angle sum identity for tangent. I don't remember it off the top of my head, but I do remember the ones for (co)sine.
By this identity, you have
So in the limit you get
The first two limits are both 1, and the single term in the last limit approaches 0 as
which agrees with the result you get from applying the chain rule.