Question

If $f'(4) = 5$, $g'(4) = 12$, $f(4)g(4) = 2$ and $g(4) = 6$, then $\left( \frac{f}{g} \right)'(4) =$

• A. $\frac{5}{36}$
• B. $\frac{11}{18}$
• C. $\frac{23}{36}$
• D. $\frac{13}{18}$
• E. $\frac{19}{36}$

Hint:

Always list out all four components ($f, g, f', g'$) before plugging them into the quotient rule. It prevents common errors like swapping the order of the numerator terms.

Answer 1

The Correct Option is D

Concept: The quotient rule for differentiation states that for two functions $f(x)$ and $g(x)$, the derivative of their quotient is: \[ \left( \frac{f}{g} \right)'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} \]

Step 1: Find the missing value of $f(4)$.
We are given $f(4)g(4) = 2$ and $g(4) = 6$. \[ f(4) \cdot 6 = 2 \quad \Rightarrow \quad f(4) = \frac{2}{6} = \frac{1}{3} \]

Step 2: Apply the quotient rule at $x = 4$.
Substitute the given values into the formula: \[ \left( \frac{f}{g} \right)'(4) = \frac{g(4)f'(4) - f(4)g'(4)}{[g(4)]^2} \] \[ \left( \frac{f}{g} \right)'(4) = \frac{(6)(5) - (\frac{1}{3})(12)}{6^2} \]

Step 3: Simplify the arithmetic expression.
\[ \text{Numerator: } 30 - 4 = 26 \] \[ \text{Denominator: } 36 \] \[ \text{Result: } \frac{26}{36} = \frac{13}{18} \]