Questions: If f(x)=(6x+2)/(7x+5), find: f'(x)=

Transcript text: If $f(x)=\frac{6 x+2}{7 x+5}$, find: \[ f^{\prime}(x)= \]

Solution

Solution Steps

To find the derivative of the function $ f(x) = \frac{6x + 2}{7x + 5} $, we will use the quotient rule. The quotient rule states that if you have a function $ f(x) = \frac{u(x)}{v(x)} $, then its derivative is given by $ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} $. Here, $ u(x) = 6x + 2 $ and $ v(x) = 7x + 5 $.

Step 1: Identify the Function Components

The given function is $ f(x) = \frac{6x + 2}{7x + 5} $. We identify the numerator as $ u(x) = 6x + 2 $ and the denominator as $ v(x) = 7x + 5 $.

Step 2: Compute the Derivatives

Calculate the derivatives of the numerator and the denominator:

The derivative of the numerator, $ u'(x) $, is $ 6 $.
The derivative of the denominator, $ v'(x) $, is $ 7 $.

Step 3: Apply the Quotient Rule

The quotient rule for derivatives states: \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \] Substituting the derivatives and the original functions, we have: \[ f'(x) = \frac{6(7x + 5) - (6x + 2)(7)}{(7x + 5)^2} \]

Step 4: Simplify the Expression

Simplifying the expression: \[ f'(x) = \frac{42x + 30 - (42x + 14)}{(7x + 5)^2} = \frac{16}{(7x + 5)^2} \]