Março 9, 2020

por Garland

Qual é a derivada de $cosx ^ 2$ ?

Responda:

$-sin2x$

Explicação:

Differentiate using the $color(blue)"chain rule"$

$color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(f(g(x)))=f'(g(x))g'(x))color(white)(a/a)|))) ........ (A)$

$color(orange)"Reminder"$

$color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(cosx)=-sinx)color(white)(a/a)|)))$
$color(blue)"-----------------------------------------------"$

$f(g(x))=cos^2x=(cosx)^2rArrf'(g(x))=2(cosx)^1=2cosx$

and $g(x)=cosxrArrg'(x)=-sinx$
$color(blue)"-----------------------------------------------"$
Substitute these values into (A)

$rArrf'(g(x))=2cosx(-sinx)=-2sinxcosx$

Using the following trig. identity to simplify.

$color(orange)"Reminder"$

$color(red)(|bar(ul(color(white)(a/a)color(black)(sin2x=2sinxcosx)color(white)(a/a)|)))$

$rarrf'(g(x))=-2sinxcosx=-sin2x$