Exercice : Fractions

Calculer les dérivées respectives des fonctions suivantes, spécifier leur ensemble de définition :

Question

\(a(x) =\frac{ x + 4}{5}\)

Indice

\(x\mapsto 1\)

\(k\mapsto 0\)

\(ku\mapsto ku'\)

\(u+v\mapsto u'+v'\)

Solution

\(a(x)=\frac{ 1}{5}x+\frac{ 4}{5}\)

\(a'(x) =\frac{1}{5}\)

\(D_a=\mathbb{R}\) et \(D_{a'}=\mathbb{R}\)

Question

\(b(x) =\frac{ x + 4}{5x}\)

Indice

\(x\mapsto 1\)

\(k\mapsto 0\)

\(ku\mapsto ku'\)

\(u+v\mapsto u'+v'\)

\(\frac{u}{v}\mapsto \frac{u'v-uv'}{v^2}\)

Solution

\(\begin{cases}u=x+4\\v=5x\end{cases}\)

\(\begin{cases}u'=1\\v'=5\end{cases}\)

\(b'(x)=\frac{ 1(5x)-5(x+4)}{(5x)^2}\)

\(\iff b'(x)=\frac{ 5x-5x-20)}{25x^2}\)

\(\iff b'(x)=\frac{ -20}{25x^2}\)

\(\iff b'(x)=\frac{ -4}{5x^2}\)

\(D_b=\mathbb{R}\backslash\{0\}\) et \(D_{b'}=\mathbb{R}\backslash\{0\}\)

Question

\(c(x) =\frac{5}{x+4}\)

Indice

\(x\mapsto 1\)

\(k\mapsto 0\)

\(ku\mapsto ku'\)

\(u+v\mapsto u'+v'\)

\(\frac{1}{u}\mapsto \frac{-u'}{u^2}\)

Solution

\(c'(x)=5 \times \frac{-1}{(x+4)^2}\)

\(\iff c'(x)=\frac{-5}{(x+4)^2}\)

\(x+4=0\iff x=-4\)

\(D_c=\mathbb{R}\backslash\{-4\}\) et \(D_{C'}=\mathbb{R}\backslash\{-4\}\)

Question

\(d(x) =\frac{3x+4}{2x+7}\)

Indice

\(x\mapsto 1\)

\(k\mapsto 0\)

\(ku\mapsto ku'\)

\(u+v\mapsto u'+v'\)

\(\frac{u}{v}\mapsto \frac{u'v-uv'}{v^2}\)

Solution

\(\begin{cases}u=3x+4\\v=2x+7\end{cases}\)

\(\begin{cases}u'=3\\v'=2\end{cases}\)

\(d'(x)=\frac{3(2x+7)-(3x+4)\times 2}{(2x+7)^2}\)

\(\iff d'(x)=\frac{6x+21-(6x+8)}{(2x+7)^2}\)

\(\iff d'(x)=\frac{6x+21-6x-8}{(2x+7)^2}\)

\(\iff d'(x)=\frac{13}{(2x+7)^2}\)

\(2x+7=0\iff 2x=-7 \iff x=-\frac{7}{2}=-3,5\)

\(D_c=\mathbb{R}\backslash\{-3,5\}\) et \(D_{C'}=\mathbb{R}\backslash\{-3,5\}\)

Question

\(e(x)=2x+3+\frac{10}{x}\)

Indice

\(x\mapsto 1\)

\(k\mapsto 0\)

\(ku\mapsto ku'\)

\(u+v\mapsto u'+v'\)

\(\frac{1}{u}\mapsto \frac{-u'}{u^2}\)

Solution

\(e(x)=2+\frac{-10}{x^2}=2-\frac{10}{x^2}\)

\(D_c=\mathbb{R}_*\) et \(D_{C'}=\mathbb{R}_*\)