If y=log⁡(2x3), find d²y/dx²

The correct option is –(3/x²).

To find the second derivative of the given function, we can simplify the expression using logarithmic identities before differentiating step-by-step with respect to $x$.

Step 1: Simplify the function using logarithm properties
The given function is:
$$y=\log \left(2x^3\right)$$
Using the product rule for logarithms, $\log \left(ab\right)=\log \left(a\right)+\log \left(b\right)$, we can rewrite this as:
$$y=\log \left(2\right)+\log \left(x^3\right)$$
Next, using the power rule for logarithms, $\log \left(u^n\right)=n\log \left(u\right)$, we further simplify the second term:
$$y=\log \left(2\right)+3\log \left(x\right)$$

Step 2: Find the first derivative
Now, we differentiate $y$ with respect to $x$. Note that $\log \left(2\right)$ is a constant value, so its derivative is $0$. The derivative of the natural logarithm $\log \left(x\right)$ is $\frac{1}{x}$:
$$\frac{dy}{dx}=\frac{d}{dx}\left[\log \right(2)+3\log (x\left)\right]$$
$$\frac{dy}{dx}=0+3\cdot \frac{1}{x}=\frac{3}{x}$$

Step 3: Find the second derivative
We differentiate the first derivative with respect to $x$ to obtain the second derivative. We can write $\frac{3}{x}$ as $3x^{-1}$ and apply the power rule:
$$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(3x^{-1}\right)$$
$$\frac{d^{2}y}{d{x}^2}=3\cdot (-1x^{-2})=-\frac{3}{x^2}$$
This simplifies to:
$$-\left(\frac{3}{x^2}\right)$$