$$ 1 \leftarrow \quad $$ Use the quotient rule to differentiate the function $$ y=\frac{4 x}{\ln (7 x)} $$.

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To differentiate the function $$ y = \frac{4x}{\ln(7x)} $$ using the quotient rule, we follow these steps:

1. **Identify the components of the quotient**:
   - Let $$ u = 4x $$ (the numerator)
   - Let $$ v = \ln(7x) $$ (the denominator)

2. **Differentiate $$ u $$ and $$ v $$**:
   - The derivative of $$ u $$ with respect to $$ x $$ is:
     $$
     u' = 4
     $$
   - The derivative of $$ v $$ with respect to $$ x $$ is:
     $$
     v' = \frac{d}{dx}(\ln(7x)) = \frac{1}{7x} \cdot 7 = \frac{1}{x}
     $$

3. **Apply the quotient rule**:
   The quotient rule states that if $$ y = \frac{u}{v} $$, then:
   $$
   y' = \frac{u'v - uv'}{v^2}
   $$
   Substituting the values we found:
   $$
   y' = \frac{(4)(\ln(7x)) - (4x)\left(\frac{1}{x}\right)}{(\ln(7x))^2}
   $$

4. **Simplify the expression**:
   $$
   y' = \frac{4\ln(7x) - 4}{(\ln(7x))^2}
   $$

Thus, the derivative of the function $$ y = \frac{4x}{\ln(7x)} $$ is:
$$
y' = \frac{4(\ln(7x) - 1)}{(\ln(7x))^2}
$$
The derivative of $$ y = \frac{4x}{\ln(7x)} $$ is $$ y' = \frac{4(\ln(7x) - 1)}{(\ln(7x))^2} $$.

Use the quotient rule to differentiate the function .

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