Answer
(a)$\frac{df}{dx}= 21.6x^{-0.7}- 8.1 x^{-1.3}$
(b)$\int \frac{df}{dx}dx =\int df=f(x)=72 x^{0.3}+27 x^{-0.3}+C$
Work Step by Step
$f(x)=72x^{0.3}+27x^{-0.3}$
(a)
$\frac{df}{dx}= \frac{d( 72x^{0.3}+27x^{-0.3} )}{dx}$
$\frac{df}{dx}= \frac{d(72x^{0.3})}{dx}+ \frac{d(27x^{ -0.3})}{dx}$
$\frac{df}{dx}= 72 \frac{d(x^{0.3})}{dx}+ 27 \frac{d(x^{ -0.3})}{dx}$
$\frac{df}{dx}= 72 (0.3)x^{0.3-1}+ 27 (-0.3)x^{-0.3-1}$
$\frac{df}{dx}= 21.6x^{-0.7}- 8.1 x^{-1.3}$
(b)
$\int \frac{df}{dx}dx =\int ( 21.6x^{-0.7}- 8.1 x^{-1.3}) dx$
$\int \frac{df}{dx}dx =21.6\int x^{-0.7} dx- 8.1 \int x^{-1.3} dx$
$\int \frac{df}{dx}dx =21.6 ( \frac{x^{ -0.7+1 }}{-0.7+1})- 8.1( \frac{x^{ -1.3+1}}{-1.3+1})$
$\int \frac{df}{dx}dx =21.6 ( \frac{x^{0.3}}{0.3})- 8.1( \frac{ x^{-0.3}}{-0.3})$
$\int \frac{df}{dx}dx =72 x^{0.3}+27 x^{-0.3}$
$\int \frac{df}{dx}dx =\int df=f(x)=72 x^{0.3}+27 x^{-0.3}+C$
