Given
[tex]y=(4x^2+7e^x)^{\frac{1}{3}}[/tex]
To find dy/dx.
Explanation:
It is given that,
[tex]y=(4x^2+7e^x)^{\frac{1}{3}}[/tex]
That implies,
[tex]\begin{gathered} \frac{dy}{dx}=\frac{d}{dx}(\left(4x^2+7e^x\right)^{\frac{1}{3}}) \\ =\frac{1}{3}\left(4x^2+7e^x\right)^{\frac{1}{3}-1}\times(4\times2x+7e^x) \\ =\frac{1}{3}\left(4x^2+7e^x\right)^{\frac{1-3}{3}}\times(8x+7e^x) \\ =\frac{1}{3}\left(4x^2+7e^x\right)^{-\frac{2}{3}}\times(8x+7e^x) \\ =\frac{8+7e^x}{3\left(4x^2+7e^x\right)^{\frac{2}{3}}} \end{gathered}[/tex]
Hence, the derivative is,
[tex]\frac{dy}{dx}=\frac{8x+7e^x}{3\left(4x^2+7e^x\right)^{\frac{2}{3}}}[/tex]
