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What is the Derivative of Cos(x)?

As in the calculation \( \frac{d}{dx} sin x \),we begin with the definition of the derivative:

\( \frac{d}{dx} cos (x) = \lim_{d \to 0} \frac{(cos(x+d)- cos(x))}{d} \)

You may remember the following angle sum formula from high school:  \( cos(a+b) = cos(a)cos(b) – sin(a) sin(b) \)

\( \frac{d}{dx} cos (x) = \lim_{d \to 0} \frac{(cos(x)cos(d) – sin(x) sin(d) – cos(x))}{d} \)

\( \frac{d}{dx} cos (x) = \lim_{d \to 0} \frac{(cos(x)cos(d) – cos(x) – sin(x) sin(d))}{d} \)

\( \frac{d}{dx} cos (x) = \lim_{d \to 0} [ \frac{cos(x)cos(d) – cos(x)}{d} + \frac{-sin(x) sin(d)}{d} ] \)

\( \frac{d}{dx} cos (x) = \lim_{d \to 0} [ \frac{cos(x)(cos(d) – 1)}{d} + \frac{-sin(x) sin(d)}{d} ] \)

\( \frac{d}{dx} cos (x) = \lim_{d \to 0} [ \frac{cos(x)(cos(d) – 1)}{d} + (-sin(x)) (\frac{ sin(d)}{d} )] \)

\( \frac{d}{dx} cos (x) = \lim_{d \to 0} cos(x) (\frac{cos(d) – 1}{d} ) + \lim_{d \to 0} (-sin(x)) (\frac{ sin(d)}{d} ) \)

Once again we use the following (unproven) facts:

\( \lim_{d \to 0} \frac{cos(d) – 1}{d}   = 0 \)

\( \lim_{d \to 0} \frac{ sin(d)}{d} = 1 \)

\( \frac{d}{dx} cos (x) = \lim_{d \to 0} cos(x) (\frac{cos(d) – 1}{d} ) + \lim_{d \to 0} (-sin(x)) (\frac{ sin(d)}{d} ) \)

\( \frac{d}{dx} cos (x) =  cos(x) \times 0 + (-sin(x)) \times 1 \)

\( \frac{d}{dx} cos (x) =  0 + (-sin(x))\)

\( \frac{d}{dx} cos (x) =  -sin(x) \)

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