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

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.

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.

Answer

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 )