Question

Qual é Derivada de cosx

Answer 1

Resposta:

Explicação passo-a-passo:

$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\\f(x)= cosx\\\\ \lim_{h \to 0} \frac{cos(x+h)-cosx}{h}\\cos(a+b)= cos(a)*cos(b)-sen(a)*sen(b)\\\\ \lim_{h \to 0} \dfrac{cosx*cosh-senx*senh-cosx}{h}\\\\ \lim_{h \to 0} \dfrac{cosx(cosh-1)-senx*senh}{h}\\ \lim_{h \to 0} \dfrac{cosx(cosh-1)}{h}- \lim_{h \to 0} \dfrac{senx*senh}{h}\\\\cosx* \lim_{h \to 0} \dfrac{cosh-1}{h}-senx* \lim_{h \to 0} \dfrac{senh}{h}\\\\lembre \ \lim_{x \to 0} \dfrac{senx}{x} = 1\\ \lim_{h \to 0} \dfrac{senh}{h}=1$

$cosx*\lim_{h \to 0} \dfrac{cosh-1}{h}-senx*1\\\\cosx*\lim_{h \to 0} \dfrac{cosh-1}{h}-senx\\\\cosx*\lim_{h \to 0} \dfrac{cosh-1}{h}*\dfrac{cosh+1}{cosh+1}-senx\\\\cosx*\lim_{h \to 0} \dfrac{cos^2h-1}{h(cosh+1)}-senx\\\\sen^2x+cos^2x=1 => sen^2x=1-cos^2x => -sen^2x= -1+cos^2x= cos^2x-1\\\\\\logo \ \\cosx* \lim_{h \to 0} \dfrac{-sen^2h}{h*(cosh+1)}$

$cosx*(\lim_{h \to 0} \dfrac{senh}{h}*\lim_{h \to 0} \dfrac{senh}{cos^2h+1})-senx\\\\\lim_{h \to 0} \dfrac{senh}{h}=1 \ limite \ fundamental \\\\\cosx(1*\dfrac{sen(0)}{cos(0)+1})-senx\\sen(0)=0 \ e \ cos(0)=1\\\\cosx*\dfrac{0}{2} -senx\\0-senx\\senx\\\\Logo\\se \y=cosx \ entao \ y'= -senx$