If y = 2ˣ3²ˣ⁻¹ then dy/dx is equal to dx

The correct answer is y (log 18).

Let us understand how to find the derivative of the given function step-by-step.

The given function is:
$$y=2^x\cdot 3^{2x-1}$$

First, we can simplify the expression for $y$ using the laws of exponents.
Recall that $a^{b-c}=\frac{a^b}{a^c}$. Therefore, we can write:
$$3^{2x-1}=\frac{3^{2x}}{3}=\frac{{\left(3^2\right)}^x}{3}=\frac{9^x}{3}$$

Substituting this back into the expression for $y$, we get:
$$y=2^x\cdot \frac{9^x}{3}$$
Combining the terms with the same exponent $x$:
$$y=\frac{{(2\cdot 9)}^x}{3}=\frac{{18}^x}{3}$$

Now, let us take the natural logarithm (log) on both sides:
$$\log y=\log \left(\frac{{18}^x}{3}\right)$$
Using the logarithmic property $\log \left(\frac{a}{b}\right)=\log a-\log b$:
$$\log y=\log \left({18}^x\right)-\log 3$$
Using the property $\log \left(a^b\right)=b\log a$:
$$\log y=x\log 18-\log 3$$

Differentiating both sides with respect to $x$:
$$\frac{1}{y}\frac{dy}{dx}=\frac{d}{dx}(x\log 18-\log 3)$$
Since $\log 18$ is a constant coefficient and $\log 3$ is a constant whose derivative is 0:
$$\frac{1}{y}\frac{dy}{dx}=\log 18$$

Multiplying both sides by $y$ gives us the final derivative:
$$\frac{dy}{dx}=y(\log 18)$$