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What is the Derivative of Sin(x)?

we are going to share formula f derivative of sin (x) and proof .

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

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

This lets us untangle the x from the d as follows:

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

We can simplify this expression using some basic algebraic facts:

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

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

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

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

We Know that

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

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

put the Value of \( \lim_{d \to 0}  (\frac {cos(d) – 1}{d} ) =0 \) & \( \lim_{d \to 0} (\frac{sin(d)}{d} ) = 1 \)

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

=> \( \frac{d}{dx} sin (x) =   cos(x)  \)

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