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#