Question

Find the derivative using the first principle of f(x) = 3sin(2x+1)

Answer 1

$f(x) = \sin(2x + 3)$

$f(h + x) = \sin(2x + 2h + 3)$

$f′ \: x = \frac{lim}{h→0} \: \: \: \frac{f(x + h) - f(x)}{h}$

$= \frac{lim}{h→0} \: \: \frac{ \sin(2x + 2h + 3 ) - \sin(2x + 3) }{h}$

$= \frac{lim}{h→0} \: \: \frac{2 \sin(h) \cos(2x + h + 3) }{h}$

$= 2 \frac{lim}{h→0} ( \frac{ \sin(h) }{h}) \cos(2x + h + 3)$

$= 2 \cos(2x + 3)$