Exercício Resolvido de Derivadas de Funções
a)
\( \displaystyle y=x^{3} \)
\[ \bbox[#99CCFF,10px]
{f´(x)=\lim_{\Delta x\rightarrow 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}}}
\]
Temos
\( f(x+\Delta x)=(x+\Delta x)^{3} \)
e
\( f(x)=x^{3} \)
\[
\begin{align}
y´ &=\lim_{\Delta x\rightarrow 0}{\frac{(x+\Delta x)^{3}-x^{3}}{\Delta x}}=\lim_{\Delta x\rightarrow 0}{\frac{x^{3}+3x^{2}\Delta x+3x\Delta x^{2}+\Delta x^{3}-x^{3}}{\Delta x}}=\\[5pt]
&=\lim_{\Delta x\rightarrow 0}{\frac{3x^{2}\Delta x+3x\Delta x^{2}+\Delta x^{3}}{\Delta x}}=\lim_{\Delta x\rightarrow 0}{\frac{\cancel{\Delta x}(3x^{2}+3x\Delta x+\Delta x^{2})}{\cancel{\Delta x}}}=\\[5pt]
&=\lim_{\Delta x\rightarrow0}{3x^{2}+3x\Delta x+\Delta x^{2}}=\lim_{\Delta x\rightarrow 0}{3x^{2}+3x.0+0^{2}}=3x^{2}
\end{align}
\]
\[ \bbox[#FFCCCC,10px]
{y´=3x^{2}}
\]