Answer:
[tex]\dfrac{dF(x)}{dx} =20e^{4x}-6x^5e^{x^{-6}}[/tex]
Step-by-step explanation:
The derivative of [tex]F(x)[/tex] is calculated as follows:
[tex]\dfrac{dF(x)}{dx}=\dfrac{d}{dx} [(5e^{4x})+(e^{-x^6})][/tex]
[tex]\dfrac{dF(x)}{dx}=\dfrac{d}{dx} [(5e^{4x})]+\dfrac{d}{dx} [(e^{-x^6})][/tex]
[tex]\dfrac{dF(x)}{dx}=5\dfrac{d}{dx} [(e^{4x})]+\dfrac{d}{dx} [(e^{-x^6})][/tex]
using the chain rule we find that
[tex]\dfrac{d}{dx} [(e^{4x})]= \dfrac{d}{d(4x)} [(e^{4x})]+ \dfrac{d}{dx} [4x] = 4e^{4x},[/tex]
[tex]\dfrac{d}{dx} [(e^{-x^6})] = \dfrac{d}{d(-x^6)} [(e^{-x^6})]+\dfrac{d}{dx} [(-x^6})]= -6x^5e^{-x^6};[/tex]
therefore,
[tex]\dfrac{dF(x)}{dx}=5\dfrac{d}{dx} [(e^{4x})]+\dfrac{d}{dx} [(e^{-x^6})] =5(4e^{4x})-6x^5e^{x^{-6}}[/tex]
[tex]\boxed{\dfrac{dF(x)}{dx} =20e^{4x}-6x^5e^{x^{-6}}}[/tex]
