Step-by-step explanation:
You will have to use the limit defition of the derivative. Brainly does not have calculus symbols, which is very unfortunate, so I can only use my words. Take the limit as the change in 'x' approaches zero. You will have:
[tex] \frac{sin(x + {x}^{.} ) - sin(x)}{ {x}^{.} } [/tex]
If you let "x dot" equal zero as the limit requires, you will get 0/0 which is indeterminate. Break up the first term using the formula for the sine of a sum. You get:
[tex] \frac{sin(x)cos( {x}^{.} ) + sin( {x}^{.})cos(x) - sin(x) }{ {x}^{.} } [/tex]
Factor a sin(x) from the first and third term to get:
[tex] \frac{sin(x)(cos( {x}^{.}) - 1 ) + sin( {x}^{.})cos(x) }{ {x}^{.} } [/tex]
Use the linearity to break the limit into two parts:
[tex] \frac{sin(x)(cos( {x}^{.}) - 1 )}{ {x}^{.} } + \frac{sin( {x}^{.} )cos(x)}{ {x}^{.} } [/tex]
First term converges to zero, second term converges to cosine. Therefore proving the derivative of sine is cosine
