Exercício Resolvido de Derivadas de Funções
publicidade
e) \( \displaystyle y=\operatorname{sen}^{2}x \)
\[ \bbox[#99CCFF,10px]
{f´(x)=\lim_{\Delta x\rightarrow 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}}}
\]
Temos
\( f(x+\Delta x)=\operatorname{sen}^{2}(x+\Delta x) \)
e
\( f(x)=\operatorname{sen}^{2}x \)
\[
y'=\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}^{2}(x+\Delta x)-\operatorname{sen}^{2}(x)}{\Delta x}}
\]
Lembrando da Trigonometria
\[ \bbox[#99CCFF,10px]
{\operatorname{sen}(a+b)=\operatorname{sen}a\cos b+\operatorname{sen}b\cos a}
\]
\[
y'=\lim_{\Delta x\rightarrow 0}{\frac{(\operatorname{sen}x\cos\Delta x+\operatorname{sen}\Delta x\cos x)^{2}-\operatorname{sen}^{2}(x)}{\Delta x}}
\]
Lembrando dos Produtos Notáveis
\[ \bbox[#99CCFF,10px]
{(a+b)^{2}=a^{2}+2ab+b^{2}}
\]
\[
y'=\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}^{2}x\cos^{2}\Delta x+2\operatorname{sen}x\cos \Delta x\operatorname{sen}\Delta x\cos x+\operatorname{sen}^{2}\Delta x\cos^{2}x-\operatorname{sen}^{2}x}{\Delta x}}
\]
colocando −sen2x em evidência
\[
y'=\lim_{\Delta x\rightarrow 0}{\frac{-\operatorname{sen}^{2}x(1-\cos^{2}\Delta x)+2\operatorname{sen}x\cos \Delta x\operatorname{sen}\Delta x\cos x+\operatorname{sen}^{2}\Delta x\cos^{2}x}{\Delta x}}
\]
usando
\[
\begin{gather}
\operatorname{sen}^{2}x+\cos^{2}x=1\\
\operatorname{sen}^{2}x=1-\cos^{2}x
\end{gather}
\]
\[
y'=\lim_{\Delta x\rightarrow 0}{\frac{-\operatorname{sen}^{2}x\operatorname{sen}^{2}\Delta x+2\operatorname{sen}x\cos \Delta x\operatorname{sen}\Delta x\cos x+\operatorname{sen}^{2}\Delta x\cos^{2}x}{\Delta x}}
\]
o limite da soma é a soma dos limites
\[
\begin{align}
y'=\lim_{\Delta x\rightarrow 0} &{\frac{-\operatorname{sen}^{2}x\operatorname{sen}^{2}\Delta x}{\Delta x}}+\lim_{\Delta x\rightarrow 0}{\frac{2\operatorname{sen}x\cos \Delta x\operatorname{sen}\Delta x\cos x}{\Delta x}}+\\
& +\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}^{2}\Delta x\cos^{2}x}{\Delta x}}
\end{align}
\]
no primeiro e no terceiro termos multiplicamos o numerador e o denominador por Δx e colocamos
os termos que não dependem de Δx para fora dos limites
\[
\begin{align}
y'=-\operatorname{sen}^{2}x\; &\lim_{\Delta x\rightarrow 0} {\frac{\operatorname{sen}^{2}\Delta x}{\Delta x}.\frac{\Delta x}{\Delta x}}+2\operatorname{sen}x\cos x\lim_{\Delta x\rightarrow 0}{\frac{\cos \Delta x\operatorname{sen}\Delta x}{\Delta x}}+\\
& +\cos^{2}x\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}^{2}\Delta x}{\Delta x}.\frac{\Delta x}{\Delta x}}
\end{align}
\]
o limite do produto é o produto dos limites
\[
\begin{align}
y'= & -\operatorname{sen}^{2}x\;\left(\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}^{2}\Delta x}{\Delta x^{2}}\Delta x}\right)+2\operatorname{sen}x\cos x\left(\lim_{\Delta x\rightarrow 0}{\cos \Delta x}\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}\Delta x}{\Delta x}}\right)+\\
& +\cos^{2}x\left(\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}^{2}\Delta x}{\Delta x^{2}}\Delta x}\right)\\[5pt]
y'= & -\operatorname{sen}^{2}x\left(\lim_{\Delta x\rightarrow 0} {\frac{\operatorname{sen}\Delta x}{\Delta x}}\right)^{2}\left(\lim_{\Delta x\rightarrow 0}{\Delta x}\right)+2\operatorname{sen}x\cos x\left(\lim_{\Delta x\rightarrow 0}{\cos \Delta x}\right)\left(\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}\Delta x}{\Delta x}}\right)+\\
& +\cos^{2}x\left(\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}\Delta x}{\Delta x}}\right)^{2}\left(\lim_{\Delta x\rightarrow 0}{\Delta x}\right)
\end{align}
\]
usando o Limite Fundamental
\[ \bbox[#99CCFF,10px]
{\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}x}{x}}=1}
\]
\[
\begin{align}
y'= &-\operatorname{sen}^{2}x{\underbrace{\left(\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}\Delta x}{\Delta x}}\right)}_{1}}^{2}\underbrace{\left(\lim_{\Delta x\rightarrow 0}{0}\right)}_{0}+2\operatorname{sen}x\cos x\underbrace{\left(\lim_{\Delta x\rightarrow 0}{\cos 0}\right)}_{1}\underbrace{\left(\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}\Delta x}{\Delta x}}\right)}_{1}+\\
& +\cos^{2}x{\underbrace{\left(\lim_{\Delta x\rightarrow 0}{\frac{\operatorname{sen}\Delta x}{\Delta x}}\right)}_{1}}^{2}\underbrace{\left(\lim_{\Delta x\rightarrow 0}{0}\right)}_{0}\\[5pt]
& \qquad \qquad \qquad \qquad \qquad \qquad \quad y'=2\operatorname{sen}x\cos x
\end{align}
\]
\[ \bbox[#FFCCCC,10px]
{y'=2\operatorname{sen}x\cos x}
\]
publicidade
Fisicaexe - Exercícios Resolvidos de Física de Elcio Brandani Mondadori está licenciado com uma Licença Creative Commons - Atribuição-NãoComercial-Compartilha Igual 4.0 Internacional .