Difference between revisions of "Derivative of cosine"

From specialfunctionswiki
Jump to: navigation, search
(Proof)
Line 17: Line 17:
  
 
==References==
 
==References==
 +
*{{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Derivative of sine|next=Derivative of tangent}}: $4.3.106$
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Proven]]
 
[[Category:Proven]]

Revision as of 02:47, 5 January 2017

Theorem

The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}x} \cos(x) = -\sin(x),$$ where $\cos$ denotes the cosine and $\sin$ denotes the sine.

Proof

From the definition of cosine, $$\cos(z) = \dfrac{e^{iz}+e^{-iz}}{2},$$ and so using the derivative of the exponential function, the linear property of the derivative, the chain rule, the reciprocal of i, and the definition of the sine function, $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \cos(z) &= \dfrac{1}{2} \left[ \dfrac{\mathrm{d}}{\mathrm{d}z} [e^{iz}] + \dfrac{\mathrm{d}}{\mathrm{d}z}[e^{-iz}] \right] \\ &= \dfrac{1}{2} \left[ ie^{iz} - ie^{-iz} \right] \\ &= -\dfrac{e^{iz}-e^{-iz}}{2i} \\ &= -\sin(z), \end{array}$$ as was to be shown. █

References