a. Using logarithmic differentiation, determine $\frac{d y}{d x}$ if $y=\frac{e^{x^{2}+1} 3^{x}}{\cos x}$.

Step 1 ：Take the natural logarithm of both sides of the equation: \(\ln y = \ln\left(\frac{e^{x^{2}+1} 3^{x}}{\cos x}\right)\)

Step 2 ：Simplify the equation using properties of logarithms: \(\ln y = (x^2 + 1) + x\ln 3 - \ln(\cos x)\)

Step 3 ：Differentiate both sides with respect to x: \(\frac{1}{y}\frac{dy}{dx} = 2x + \ln 3 - \frac{\sin x}{\cos x}\)

Step 4 ：Multiply both sides by y to find the derivative of y with respect to x: \(\frac{dy}{dx} = y\left(2x + \ln 3 - \frac{\sin x}{\cos x}\right)\)

Step 5 ：Substitute the original expression for y back into the equation: \(\boxed{\frac{dy}{dx} = \frac{3^x e^{x^2 + 1}(2x - \frac{\sin x}{\cos x} + \ln 3)}{\cos x}}\)