probar con h≠0 ((sin(x+h)−sin(x))/h)=((sin(h/2))/(h/2))cos(x+(h/2))


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Question Number 59006 by arcana last updated on 03/May/19

$p r o b a r c o n h \neq 0$ $\frac{s i n (x + h) - s i n (x)}{h} = \frac{s i n (h / 2)}{h / 2} c o s (x + \frac{h}{2})$
Commented by arcana last updated on 25/May/19

$u s a n d o 2 s i n (x) c o s (y) = s i n (x + y) - s i n (y - x)$ $t o m a r x = \frac{h}{2}, y = x + \frac{h}{2}$ $l u e g o 2 s i n (h / 2) c o s (x + h / 2) = s i n (x + h) - s i n (x)$ $\frac{2}{h} s i n (h / 2) c o s (x + h / 2) = \frac{1}{h} [s i n (x + h) - s i n (x)]$ $\frac{s i n (h / 2) c o s (x + h / 2)}{h / 2} = \frac{s i n (x + h) - s i n (x)}{h}$ $\frac{d s e n}{d x} (x) = \underset{h \to 0}{l i m} \frac{s i n (x + h) - s i n (x)}{h} = \underset{h \to 0}{l i m} \frac{s i n (h / 2) c o s (x + h / 2)}{h / 2}$ $\frac{d s e n}{d x} (x) = \underset{h \to 0}{l i m} \frac{s i n (h / 2)}{h / 2} \underset{h \to 0}{l i m} c o s (x + h / 2)$ $\frac{d s e n}{d x} (x) = \underset{h \to 0}{l i m c o s} (x + h / 2) = c o s (x)$
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