Question

i need help question

Answer 1

Solution

Given

$\begin{gathered} y=\sqrt{x}e^{x^3}(x^2+1)^6 \\ \\ \Rightarrow\ln(y)=\ln(\sqrt{x})+\ln(e^{x^3})+\ln((x^2+1)^6) \\ \\ \Rightarrow\frac{1}{y}\frac{dy}{dx}=\frac{1}{2x}+3x^2+6\frac{2x}{(x^2+1)} \\ \\ \Rightarrow\frac{dy}{dx}=y(\frac{1}{2x}+3x^2+\frac{12x}{(x^2+1)}) \end{gathered}$

=>

$y^{\prime}=\sqrt{x}e^{x^3}(x^2+1)^6(\frac{1}{2x}+3x^2+\frac{12x}{(x^2+1)})$