1
Aqui apresentamos um exemplo resolvido passo a passo de regras básicas de diferenciação. Esta solução foi gerada automaticamente pela nossa calculadora inteligente:
$\frac{d}{dx}\left(\frac{x^2+3x+1}{x^2+2x+2}\right)^2$
2
Aplicamos a regra: $\left(\frac{a}{b}\right)^n$$=\frac{a^n}{b^n}$, onde $a=x^2+3x+1$, $b=x^2+2x+2$ e $n=2$
$\frac{d}{dx}\left(\frac{\left(x^2+3x+1\right)^2}{\left(x^2+2x+2\right)^2}\right)$
Passos
Aplicamos a regra: $\frac{d}{dx}\left(\frac{a}{b}\right)$$=\frac{\frac{d}{dx}\left(a\right)b-a\frac{d}{dx}\left(b\right)}{b^2}$, onde $a=\left(x^2+3x+1\right)^2$ e $b=\left(x^2+2x+2\right)^2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(\left(x^2+2x+2\right)^2\right)^2}$
Simplifique $\left(\left(x^2+2x+2\right)^2\right)^2$ aplicando a potência de uma potência: $\left(a^m\right)^n=a^{m\cdot n}$. Na expressão, $m$ é igual a $2$ e $n$ é igual a $2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{4}}$
3
Aplicamos a regra: $\frac{d}{dx}\left(\frac{a}{b}\right)$$=\frac{\frac{d}{dx}\left(a\right)b-a\frac{d}{dx}\left(b\right)}{b^2}$, onde $a=\left(x^2+3x+1\right)^2$ e $b=\left(x^2+2x+2\right)^2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(\left(x^2+2x+2\right)^2\right)^2}$
Explique melhor esta etapa
Passos
Simplifique $\left(\left(x^2+2x+2\right)^2\right)^2$ aplicando a potência de uma potência: $\left(a^m\right)^n=a^{m\cdot n}$. Na expressão, $m$ é igual a $2$ e $n$ é igual a $2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{2\cdot 2}}$
Simplifique $\left(\left(x^2+2x+2\right)^2\right)^2$ aplicando a potência de uma potência: $\left(a^m\right)^n=a^{m\cdot n}$. Na expressão, $m$ é igual a $2$ e $n$ é igual a $2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{2\cdot 2}}$
Aplicamos a regra: $ab$$=ab$, onde $ab=2\cdot 2$, $a=2$ e $b=2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{4}}$
Aplicamos a regra: $ab$$=ab$, onde $ab=2\cdot 2$, $a=2$ e $b=2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{4}}$
4
Simplifique $\left(\left(x^2+2x+2\right)^2\right)^2$ aplicando a potência de uma potência: $\left(a^m\right)^n=a^{m\cdot n}$. Na expressão, $m$ é igual a $2$ e $n$ é igual a $2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{4}}$
Explique melhor esta etapa
5
Aplicamos a regra: $\frac{d}{dx}\left(x^a\right)$$=ax^{\left(a-1\right)}\frac{d}{dx}\left(x\right)$, onde $a=2$ e $x=x^2+3x+1$
$\frac{2\left(x^2+3x+1\right)\frac{d}{dx}\left(x^2+3x+1\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{4}}$
6
Aplicamos a regra: $\frac{d}{dx}\left(x^a\right)$$=ax^{\left(a-1\right)}\frac{d}{dx}\left(x\right)$, onde $a=2$ e $x=x^2+2x+2$
$\frac{2\left(x^2+3x+1\right)\frac{d}{dx}\left(x^2+3x+1\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\frac{d}{dx}\left(x^2+2x+2\right)}{\left(x^2+2x+2\right)^{4}}$
7
A derivada da soma de duas ou mais funções é equivalente à soma das derivadas de cada função separadamente
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(1\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\frac{d}{dx}\left(x^2+2x+2\right)}{\left(x^2+2x+2\right)^{4}}$
8
A derivada da soma de duas ou mais funções é equivalente à soma das derivadas de cada função separadamente
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(1\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(2\right)\right)}{\left(x^2+2x+2\right)^{4}}$
9
Aplicamos a regra: $\frac{d}{dx}\left(c\right)$$=0$, onde $c=1$
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(3x\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(2\right)\right)}{\left(x^2+2x+2\right)^{4}}$
10
Aplicamos a regra: $\frac{d}{dx}\left(c\right)$$=0$, onde $c=2$
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(3x\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)}{\left(x^2+2x+2\right)^{4}}$
Passos
Aplicamos a regra: $\frac{d}{dx}\left(cx\right)$$=c\frac{d}{dx}\left(x\right)$
$3\frac{d}{dx}\left(x\right)$
Aplicamos a regra: $\frac{d}{dx}\left(x\right)$$=1$
$3$
11
Aplicamos a regra: $\frac{d}{dx}\left(nx\right)$$=n\frac{d}{dx}\left(x\right)$, onde $n=3$
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+3\frac{d}{dx}\left(x\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)}{\left(x^2+2x+2\right)^{4}}$
Explique melhor esta etapa
Passos
Aplicamos a regra: $\frac{d}{dx}\left(cx\right)$$=c\frac{d}{dx}\left(x\right)$
$2\frac{d}{dx}\left(x\right)$
Aplicamos a regra: $\frac{d}{dx}\left(x\right)$$=1$
$2$
12
Aplicamos a regra: $\frac{d}{dx}\left(nx\right)$$=n\frac{d}{dx}\left(x\right)$, onde $n=2$
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+3\frac{d}{dx}\left(x\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+2\frac{d}{dx}\left(x\right)\right)}{\left(x^2+2x+2\right)^{4}}$
Explique melhor esta etapa
Passos
Aplicamos a regra: $\frac{d}{dx}\left(x\right)$$=1$
$3\cdot 1$
Aplicamos a regra: $1x$$=x$, onde $x=3$
$3$
13
Aplicamos a regra: $\frac{d}{dx}\left(x\right)$$=1$
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+3\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+2\frac{d}{dx}\left(x\right)\right)}{\left(x^2+2x+2\right)^{4}}$
Explique melhor esta etapa
Passos
Aplicamos a regra: $\frac{d}{dx}\left(x\right)$$=1$
$3\cdot 1$
Aplicamos a regra: $1x$$=x$, onde $x=3$
$3$
Aplicamos a regra: $\frac{d}{dx}\left(x\right)$$=1$
$2\cdot 1$
Aplicamos a regra: $1x$$=x$, onde $x=2$
$2$
14
Aplicamos a regra: $\frac{d}{dx}\left(x\right)$$=1$
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+3\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+2\right)}{\left(x^2+2x+2\right)^{4}}$
Explique melhor esta etapa
Passos
Aplicamos a regra: $\frac{d}{dx}\left(x^a\right)$$=ax^{\left(a-1\right)}$, onde $a=2$
$2x^{\left(2-1\right)}$
Aplicamos a regra: $a+b$$=a+b$, onde $a=2$, $b=-1$ e $a+b=2-1$
$2x$
15
Aplicamos a regra: $\frac{d}{dx}\left(x^a\right)$$=ax^{\left(a-1\right)}$, onde $a=2$
$\frac{2\left(x^2+3x+1\right)\left(2x+3\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(2x+2\right)}{\left(x^2+2x+2\right)^{4}}$
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Resposta final para o problema
$\frac{2\left(x^2+3x+1\right)\left(2x+3\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(2x+2\right)}{\left(x^2+2x+2\right)^{4}}$