Duas regras de derivação
Derivadas: Aleph 11 - Volume 2 Pág. 73 Ex. 2
Determine regras de derivação que permitam calcular facilmente derivadas de funções do tipo:
\[\begin{array}{*{20}{c}}
{f\left( x \right) = \frac{k}{{x – a}}}&{}&{}&{g\left( x \right) = \frac{k}{{{x^2}}}}
\end{array}\]
\[\begin{array}{*{20}{c}}
{f\left( x \right) = \frac{k}{{x – a}}}&{}&{}&{g\left( x \right) = \frac{k}{{{x^2}}}}
\end{array}\]
Seja $k$ constante e ${x_0} \in \mathbb{R}\backslash \left\{ a \right\}$.
\[\begin{array}{*{20}{l}}
{f’\left( {{x_0}} \right)}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {{x_0} + h} \right) – f\left( {{x_0}} \right)}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{\frac{k}{{{x_0} + h – a}} – \frac{k}{{{x_0} – a}}}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{\frac{{k{x_0} – ka – k{x_0} – kh + ka}}{{\left( {{x_0} + h – a} \right)\left( {{x_0} – a} \right)}}}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{ – kh}}{{h\left( {{x_0} + h – a} \right)\left( {{x_0} – a} \right)}}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{ – k}}{{\left( {{x_0} + h – a} \right)\left( {{x_0} – a} \right)}}} \\
{}& = &{\frac{{ – k}}{{{{\left( {{x_0} – a} \right)}^2}}}}
\end{array}\]
Logo:\[\begin{array}{*{20}{c}}
{\begin{array}{*{20}{l}}
{f:}&{\mathbb{R}\backslash \left\{ a \right\} \to \mathbb{R}} \\
{}&{x \to \frac{k}{{x – a}}}
\end{array}}&{}&{}&{\begin{array}{*{20}{l}}
{f’:}&{\mathbb{R}\backslash \left\{ a \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{ – k}}{{{{\left( {x – a} \right)}^2}}}}
\end{array}}
\end{array}\]
Seja $k$ constante e ${x_0} \in \mathbb{R}\backslash \left\{ 0 \right\}$.
\[\begin{array}{*{20}{l}}
{g’\left( {{x_0}} \right)}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{g\left( {{x_0} + h} \right) – g\left( {{x_0}} \right)}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{\frac{k}{{{{\left( {{x_0} + h} \right)}^2}}} – \frac{k}{{{x_0}^2}}}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{\frac{{k{x_0}^2 – k{x_0}^2 – 2kh{x_0} – k{h^2}}}{{{x_0}^2{{\left( {{x_0} + h} \right)}^2}}}}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{ – 2kh{x_0} – k{h^2}}}{{h{x_0}^2{{\left( {{x_0} + h} \right)}^2}}}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{ – 2k{x_0} – kh}}{{{x_0}^2{{\left( {{x_0} + h} \right)}^2}}}} \\
{}& = &{\frac{{ – 2k{x_0}}}{{{x_0}^4}}} \\
{}& = &{\frac{{ – 2k}}{{{x_0}^3}}}
\end{array}\]
Logo:
\[\begin{array}{*{20}{c}}
{\begin{array}{*{20}{l}}
{g:}&{\mathbb{R}\backslash \left\{ 0 \right\} \to \mathbb{R}} \\
{}&{x \to \frac{k}{{{x^2}}}}
\end{array}}&{}&{}&{\begin{array}{*{20}{l}}
{g’:}&{\mathbb{R}\backslash \left\{ 0 \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{ – 2k}}{{{x^3}}}}
\end{array}}
\end{array}\]