Find from first principles the derivative of cos x

Question 1

Question 2

Answer:

Please see the explanation.

Explanation:

Let

$f\left(x\right)=cosx$

By the first principle

$f\:'\left(x\right)=\lim _(h\to 0)\left((f\left(x+h\right)-f\left(x\right))/(h)\right)$

$=\lim _(h\to 0)\left((cos\:\left(x+h\right)-cos\:x)/(h)\right)$

$=\lim _(h\to 0)\left[(cos\:x\:cos\:h-sin\:x\:sin\:h\:-\:cos\:x)/(h)\right]$

$=\lim _(h\to 0)\left[(-cos\:x\left(1-cos\:h\right)-sin\:x\:sin\:h\:)/(h)\right]$

$=\lim _(h\to 0)\left[(-cos\:x\left(1-cos\:h\right)\:)/(h)-(sin\:x\:sin\:h)/(h)\right]$

$=-cosx\:\left(\lim \:_(h\to \:0\:)(1-cos\:h)/(h)\right)-sin\:x\:\lim \:\:_(h\to \:\:0)\:\left((sin\:h)/(h)\right)$

$=-cosx\:\left(0\right)-sinx\left(1\right)$

$=-sin\:x$

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