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Mathematics LibreTexts

4.2: The Derivative of 1/sin x

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What about the derivative of the sine function? The rules for derivatives that we have are no help, since \sin x is not an algebraic function. We need to return to the definition of the derivative, set up a limit, and try to compute it. Here's the definition:

{d\over dx}\sin x = \lim_{\Delta x\to0} {\sin(x+\Delta x)-\sin x \over \Delta x}. \nonumber

Using some trigonometric identities, we can make a little progress on the quotient:

\eqalign{ {\sin(x+\Delta x)-\sin x \over \Delta x}&={\sin x \cos \Delta x + \sin \Delta x \cos x - \sin x\over \Delta x}\cr& =\sin x{\cos \Delta x - 1\over \Delta x}+\cos x{\sin\Delta x\over \Delta x}.\cr } \nonumber

This isolates the difficult bits in the two limits

\lim_{\Delta x\to0}{\cos \Delta x - 1\over \Delta x}\quad\hbox{and}\quad \lim_{\Delta x\to0} {\sin\Delta x\over \Delta x}. \nonumber

Here we get a little lucky: it turns out that once we know the second limit the first is quite easy. The second is quite tricky, however. Indeed, it is the hardest limit we will actually compute, and we devote a section to it.

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This page titled 4.2: The Derivative of 1/sin x is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by David Guichard via source content that was edited to the style and standards of the LibreTexts platform.

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