Exercice : Polynômes

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

Question

\(b(x)=2x^4 − 3x^2 + 7\)

Indice

\(x^n\mapsto nx^{n-1}\)

\(x \mapsto 1\)

\(k\mapsto 0\)

\(ku\mapsto ku'\)

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

Solution

\(b'(x)=2\times 4x^3-3 \times 2x\)

\(b'(x)=8x^3-6x\)

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

Question

\(c(x)=x^{3}+x^{2}+x+1\)

Indice

\(x^n\mapsto nx^{n-1}\)

\(x \mapsto 1\)

\(k\mapsto 0\)

\(ku\mapsto ku'\)

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

Solution

\(c'(x)=3 x^2+2x+1\)

\(D_c=\mathbb{R}\) et \(D_{c'}=\mathbb{R}\)

Question

\(d(x)=3x^{2}-5x+4.\)

Indice

\(x^n\mapsto nx^{n-1}\)

\(x \mapsto 1\)

\(k\mapsto 0\)

\(ku\mapsto ku'\)

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

Solution

\(d'(x)=3 \times 2x-5 \times 1\)

\(\iff f'(x)=6x-5\)

\(D_d=\mathbb{R}\) et \(D_{d'}=\mathbb{R}\)

Question

\(e(x)=-2x^{3}+3x^{2}+9.\)

Indice

\(x^n\mapsto nx^{n-1}\)

\(x \mapsto 1\)

\(k\mapsto 0\)

\(ku\mapsto ku'\)

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

Solution

\(e'(x)=-2 \times 3x^2+3 \times 2x\)

\(\iff e'(x)=-6x^2+6x\)

\(D_e=\mathbb{R}\) et \(D_{e'}=\mathbb{R}\)

Question

\(f(x)=3x^{2}+5x\)

Indice

\(x^n\mapsto nx^{n-1}\)

\(x \mapsto 1\)

\(ku\mapsto ku'\)

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

Solution

\(f'(x)=3\times 2x+5 \times 1\)

\(\iff f'(x)=6x+5\)

\(D_f=\mathbb{R}\) et \(D_{f'}=\mathbb{R}\)

Question

\(g(x)=\frac{x^{2}}{8}+9\)

Indice

\(x^n\mapsto nx^{n-1}\)

\(x \mapsto 1\)

\(k\mapsto 0\)

\(ku\mapsto ku'\)

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

Solution

\(g(x)=\frac{x^{2}}{8}+9\)

\(\iff g(x)=\frac{1}{8}x^{2}+9\)

\(g'(x)=\frac{1}{8}\times 2x\)

\(\iff g'(x)=\frac{2}{8}x\)

\(\iff g'(x)=\frac{1}{4}x\)

\(D_g=\mathbb{R}\) et \(D_{g'}=\mathbb{R}\)

Question

\(h(x)=5x^2-8x+2\)

Indice

\(x \mapsto 1\)

\(ku\mapsto ku'\)

\(k\mapsto 0\)

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

\(x^n\mapsto nx^{n-1}\)

Solution

\(h'(x)=5\times 2x-8\times 1+0\)

\(\iff h'(x)=10x-8\)

\(D_h=\mathbb{R}\) et \(D_{h'}=\mathbb{R}\)

Question

\(i(x)=-4x^3+7x+9\)

Indice

\(x \mapsto 1\)

\(ku\mapsto ku'\)

\(k\mapsto 0\)

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

\(x^n\mapsto nx^{n-1}\)

Solution

\(i'(x)=-4\times 3x^2+7\times 1+0\)

\(\iff i'(x)=-12x^2+7\)

\(D_i=\mathbb{R}\) et \(D_{i'}=\mathbb{R}\)

Question

\(j(x)=2x^{3}+\dfrac{x^{2}}{4}-6x+3\)

Indice

\(x \mapsto 1\)

\(ku\mapsto ku'\)

\(k\mapsto 0\)

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

\(x^n\mapsto nx^{n-1}\)

Solution

\(j'(x)=2\times 3x^2+\dfrac{1}{4}\times 2x-6 \times 1+0\)

\(\iff j'(x)=6x^2+\dfrac{1}{2}\times x-6\)

\(D_j=\mathbb{R}\) et \(D_{j'}=\mathbb{R}\)

Question

\(k(x)=(x+2)(-3x+5)-1\)

Indice

\(x \mapsto 1\)

\(ku\mapsto ku'\)

\(k\mapsto 0\)

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

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

Solution

\(\begin{cases}u=x+2\\v=-3x+5\end{cases}\)

\(\begin{cases}u'=1\\v'=-3\end{cases}\)

\(k'(x)=1(-3x+5)+(-3)(x+2)\)

\(\iff k'(x)=-3x+5-3x-6\)

\(\iff h'(x)=-6x-1\)

\(D_k=\mathbb{R}\) et \(D_{k'}=\mathbb{R}\)