Question

Como obter a derivada da função:

a) f(x) = 4x^5 - 8x^4 + 2x^3


b) f(x) = - _9_ + __6_
                  x^7     x^4

Answer 1

Derivada de potência:

$\dfrac{d}{dx}(x^{n})=n\,x^{n-1}$

___________

a) $f(x)=4x^5-8x^4+2x^3$


Derivando,

$\dfrac{df}{dx}(x)=\dfrac{d}{dx}(4x^5-8x^4+2x^3)\\\\\\ \dfrac{df}{dx}(x)=4\cdot 5x^{5-1}-8\cdot 4x^{4-3}+2\cdot 3x^{3-1}\\\\ \boxed{\begin{array}{c}\dfrac{df}{dx}(x)=20x^{4}-32x^{3}+6x^{2}\end{array}}$

___________

b) $f(x)=-\,\dfrac{9}{x^7}+\dfrac{6}{x^4}$

$f(x)=-9x^{-7}+6x^{-4}$


Derivando

$\dfrac{df}{dx}(x)=\dfrac{d}{dx}(-9x^{-7}+6x^{-4})\\\\\\ \dfrac{df}{dx}(x)=-9\cdot (-7)x^{-7-1}+6\cdot (-4)x^{-4-1}\\\\\\ \dfrac{df}{dx}(x)=63x^{-8}-24x^{-5}\\\\\\ \boxed{\begin{array}{c}\dfrac{df}{dx}(x)=\dfrac{63}{x^8}-\dfrac{24}{x^5} \end{array}}$


Bons estudos! :-)