Determine the derivative of $h\left(x\right)=2x^{2\ln \left(x\right)}$. Evaluate $h^{\prime }\left(e\right)$.

Find the derivative of the function $y={\left(2x\right)}^{3\sqrt{x-2}}$.

Use logarithmic differentiation to find the derivative of the function $f\left(x\right)=\frac{x^5{\sin }^{2}x}{\sqrt{x+7^{}}}$.

Find the derivative of the given function by using logarithmic differentiation or by using the property $b^x=e^{x\ln b}$.

$y={(7x^2+5)}^x$