Qual é a derivada de: f(x) = 4x³ + 2x² + 2/(x³)

 Qual é a derivada de 

$�(�)=4{�}^3+2{�}^2+\frac{2}{{�}^3}$?

Solução

Passo 1: Começamos convertendo a expressão racional para uma expressão com um expoente numérico:

$�(�)=4{�}^3+2{�}^2+2{�}^{-3}$

Passo 2: Usando a regra da potência para os três termos, temos:

$�(�)=4{�}^3+2{�}^2+2{�}^{-3}$

${�}^{\mathrm{\prime }}(�)=12{�}^2+4�-6{�}^{-4}$

Passo 3: Usamos as leis dos expoentes novamente para escrever da seguinte forma:

$${\mathrm{<span\; style="font-family:\; times;\; text-align:\; justify;"><span\; style="color:\; \#274e13;\; font-size:\; large;">�</span></span>}}^{\mathrm{<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">\prime </span></span>}}<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">(</span></span>\mathrm{<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">�</span></span>}<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">)</span></span><span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">=</span></span>\mathrm{<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">12</span></span>}{\mathrm{<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">�</span></span>}}^{\mathrm{<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">2</span></span>}}<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">+</span></span>\mathrm{<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">4</span></span>}\mathrm{<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">�</span></span>}<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">-</span></span>\frac{\mathrm{<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">6</span></span>}}{{\mathrm{<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">�</span></span>}}^{\mathrm{<span\; style="font-family:\; times;"><span\; style="color:\; \#274e13;\; font-size:\; large;">4</span></span>}}}$$