Question

ENCONTRAR A DERIVADA DE F(X) = (X-)^3 (3X^2 -8X -12)

Answer 1

$f(x)=(x^{-5})^{3}\cdot(3x^{2}-8x-12)$

Utilizando a propriedade de potências $(a^{m})^{n}=a^{m\cdot n}$:

$f(x)=x^{(-5)\cdot3}\cdot(3x^{2}-8x-12)\\\\f(x)=x^{-15}\cdot(3x^{2}-8x-12)$

Aplicando a propriedade distributiva:

$f(x)=x^{-15}\cdot(3x^{2}-8x-12)\\\\f(x)=3x^{2}\cdot x^{-15}-8x\cdot x^{-15}-12x^{-15}\\\\f(x)=3x^{2-15}-8x^{1-15}-12x^{-15}\\\\f(x)=3x^{-13}-8x^{-14}-12x^{-15}$

Agora, vamos usar as propriedades de derivação para derivar f(x):

$\boxed{\boxed{\dfrac{d}{dx}[f(x)\pm g(x)\pm...\pm z(x)]=f'(x)\pm g'(x)\pm...\pm z'(x)}}\\\\\\\boxed{\boxed{\dfrac{d}{dx}[c\cdot f(x)]=c\dfrac{d}{dx}f(x)=cf'(x)}}\\\\\\\boxed{\boxed{\dfrac{d}{dx}(x^{n})=n\cdot x^{n-1}}}$
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Derivando f(x):

$f'(x)=\frac{d}{dx}(3x^{-13})-\frac{d}{dx}(8x^{-14})-\frac{d}{dx}(12x^{-15})\\\\f'(x)=3\frac{d}{dx}(x^{-13})-8\frac{d}{dx}(x^{-14})-12\frac{d}{dx}(x^{-15})\\\\f'(x)=3\cdot(-13)x^{-13-1}-8\cdot(-14)x^{-14-1}-12\cdot(-15)x^{-15-1}\\\\\boxed{\boxed{f'(x)=-39x^{-14}+112x^{-15}+180x^{-16}}}$