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Dois jovens matemáticos discutem derivadas de produtos e produtos de
derivadas.
Confira este diálogo entre dois estudantes de cálculo (baseado em uma história real):
- Devyn
- Ei, Riley, lembra da regra da soma para derivadas?
- Riley
- Você sabe que sim.
- Devyn
- O que você acha que vai ser a “regra do produto”?
- Riley
- Vamos arriscar um palpite: \[ \dd {x} \left (f(x)\cdot g(x)\right ) = f'(x) \cdot g'(x)? \]
- Devyn
- Hmmm, vamos colocar essa teoria à prova. Vamos tentar com \[ f(x) = x^2+1\qquad \text {e}\qquad g(x) = x^3-3x \] Agora \begin{align*} f'(x)g'(x) &= (2x)(3x^2-3)\\ &= 6x^3-6x. \end{align*}
- Riley
- Por outro lado, \begin{align*} f(x)g(x) &= (x^2+1)(x^3-3x)\\ &=x^5-3x^3+x^3-3x\\ &=x^5-2x^3-3x. \end{align*}
- Devyn
- E assim, \[ \ddx \left (f(x) \cdot g(x)\right ) = 5x^4-6x^2-3. \]
- Riley
- Uau. Hmmm. Parece que o nosso palpite estava errado.
- Devyn
- Tenho a sensação de que essa tal de “regra do produto” pode ser um pouco traiçoeira.
Acima, nossos intrépidos jovens matemáticos supõem que a “regra do produto”
poderia ser: \[ \dd {x} \left (f(x)\cdot g(x)\right ) = f'(x) \cdot g'(x)? \] Isso vale em algum caso?