What is the derivative of #sinx/x#?

Answer 1
Using the quotient rule, the answer is #\frac{d}{dx}((sin(x))/x)=\frac{xcos(x)-sin(x)}{x^{2}}#
While this is technically only true for #x!=0#, an interesting thing about this example is that its discontinuity and lack of differentiability at #x=0# can be "removed".
Let #f(x)=sin(x)/x#. Use your calculator to graph this over some window near #x=0#. You'll see that the graph has no vertical asymptote, in spite of the fact that the function is undefined at #x=0#. In fact, you should see that #lim_{x->0}f(x)=1#.
Therefore, if we declare that we want #f(0)=1#, we will have created a continuous function for all #x# (a piecewise formula should be written to make this definition most clearly).
Moreover, it's also differentiable everywhere and #f'(0)=0#. This is also consistent with the fact that #lim_{x->0}\frac{x\cos(x)-sin(x)}{x^{2}}=0#, as you can check with your calculator.
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Answer 2

The derivative of sin(x)/x is (x*cos(x) - sin(x))/x^2.

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Answer from HIX Tutor

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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