Ensino Superior ⇒ (UFPB - 1970) Derivada Tópico resolvido

[ALDRINMOD]

Dada a função [tex3]y = \ell n \text{tg}^2\,x,[/tex3] o valor numérico de sua derivada primeira para [tex3]x = 45^\circ[/tex3] é:

a) 1/2.[tex3]\hspace{40pt}[/tex3] b) 1/4.[tex3]\hspace{40pt}[/tex3] c) 4.[tex3]\hspace{40pt}[/tex3] d) 2.[tex3]\hspace{40pt}[/tex3] e) 1/3.

[Karl Weierstrass]

• [tex3]y \,= \,\ell n \text{tg}\,^2\,x[/tex3]
  
  [tex3]y \, =\,2\,\cdot\, \ell n \text{tg}\,x[/tex3]
  
  [tex3]y' \, =\,2\,\cdot\, \Large\frac{(\text{tg}\,x)'}{\text{tg}\,x}\large[/tex3]
  
  [tex3]y' \, =\,2\,\cdot\, \Large\frac{\sec^2\,x}{\text{tg}\,x}\large[/tex3]
  
  [tex3]y' \, =\,2\,\cdot\, \Large\frac{1}{\cos^2\,x\,\cdot\,\Large\frac{\text{sen}\,x}{\cos\,x}\large}\large[/tex3]
  
  [tex3]y' \, =\,2\,\cdot\, \Large\frac{1}{\text{sen}\,x\,\cdot\,\cos\,x}\large[/tex3]
  
  [tex3]y' \, =\,2\,\cdot\, \Large\frac{2}{\text{sen}\,2x}\large[/tex3]
  
  [tex3]y' \, =\,\Large\frac{4}{\text{sen}\,2x}\large[/tex3]
  
  [tex3]f'(45^{\circ}) \, =\,\Large\frac{4}{\text{sen}\,90^{\circ}}\large\,=\,4.[/tex3]

[tex3]\boxed{\text{C}}[/tex3]

[ALDRINMOD]

Valeu Fera.