Questions: Find the derivative of y=ln ((3 x^2)/(sqrt(6 x+2))). a.) dy/dx=(2/x)-(1/(2(6 x+2))) b.) dy/dx=(2/x)-(3/(6 x+2)) c.) dy/dx=ln (3)+(2/x)-(1/(2(6 x+2))) d.) dy/dx=ln (3)+2 ln (x)-(1/2) ln (6 x+2)

Transcript text: Find the derivative of $y=\ln \left(\frac{3 x^{2}}{\sqrt{6 x+2}}\right)$. a.) $\frac{d y}{d x}=\frac{2}{x}-\frac{1}{2(6 x+2)}$ b.) $\frac{d y}{d x}=\frac{2}{x}-\frac{3}{6 x+2}$ c.) $\frac{d y}{d x}=\ln (3)+\frac{2}{x}-\frac{1}{2(6 x+2)}$ d.) $\frac{d y}{d x}=\ln (3)+2 \ln (x)-\frac{1}{2} \ln (6 x+2)$

Solution

Solution Steps

Step 1: Define the Function

We start with the function given in the problem: \[ y = \ln \left(\frac{3 x^{2}}{\sqrt{6 x + 2}}\right) \]

Step 2: Simplify the Logarithmic Expression

Using properties of logarithms, we can simplify the expression: \[ y = \ln(3 x^2) - \ln(\sqrt{6 x + 2}) = \ln(3) + \ln(x^2) - \frac{1}{2} \ln(6 x + 2) \] This further simplifies to: \[ y = \ln(3) + 2 \ln(x) - \frac{1}{2} \ln(6 x + 2) \]

Step 3: Differentiate Each Term

Now we differentiate each term in the simplified expression:

The derivative of $ \ln(3) $ is $ 0 $.
The derivative of $ 2 \ln(x) $ is: \[ \frac{d}{dx}[2 \ln(x)] = \frac{2}{x} \]
The derivative of $ -\frac{1}{2} \ln(6 x + 2) $ using the chain rule is: \[ -\frac{1}{2} \cdot \frac{1}{6 x + 2} \cdot 6 = -\frac{3}{6 x + 2} \]

Step 4: Combine the Derivatives

Combining the derivatives from the previous step, we have: \[ \frac{d y}{d x} = 0 + \frac{2}{x} - \frac{3}{6 x + 2} \]

Step 5: Simplify the Result

The final expression for the derivative can be simplified to: \[ \frac{d y}{d x} = \frac{2}{x} - \frac{3}{6 x + 2} \]