Question

the derivative of $y = 7ln\frac{1}{2 - 3x}$ is $\frac{dy}{dx}=\frac{2}{2 - 3x}$. find the value of c. give an exact answer as an integer.

Explanation:

Step1: Differentiate \( y = 7\ln\frac{1}{2 - 3x} \)

Use the chain - rule. First, rewrite \( y = 7\ln(2 - 3x)^{-1}=-7\ln(2 - 3x) \).
The derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{u'}{u} \), where \( u = 2 - 3x \) and \( u'=-3 \).
So, \( y'=\frac{dy}{dx}=-7\times\frac{-3}{2 - 3x}=\frac{21}{2 - 3x} \).

Step2: Compare with \( \frac{1}{\frac{dy}{dx}}=\frac{2 - 3x}{c} \)

Since \( \frac{dy}{dx}=\frac{21}{2 - 3x} \), then \( \frac{1}{\frac{dy}{dx}}=\frac{2 - 3x}{21} \).
Comparing \( \frac{2 - 3x}{21} \) with \( \frac{2 - 3x}{c} \).

Answer:

\( 21 \)