Question: Suppose that $f(x)$ and $g(x)$ are differentiable functions such that $g(0) = 5$, $g'(0) = 6$, $f(0) = 1$, and $f'(0) = 9$. Find
$h'(0)$ when $h(x) = \frac{f(x)}{g(x)}$.
Which of the following expressions could be used to compute $h'(0)$?
A. $\frac{g(0)f'(0) - f(0)g'(0)}{[g(0)]^2}$
B. $\frac{f(0)g'(0) - g(0)f'(0)}{[g(0)]^2}$
C. $f'(0)g(0) - f(0)g'(0)$
D. $f(0)g'(0) - f'(0)g(0)$
1. **State the problem:** We want to find the derivative $h'(0)$ of the function $$h(x) = \frac{f(x)}{g(x)}$$ at $x=0$, given values for $f(0)$, $f'(0)$, $g(0)$, and $g'(0)$.
2. **Recall the quotient rule:** For differentiable functions $f$ and $g$, the derivative of their quotient is
$$h'(x) = \left(\frac{f}{g}\right)'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}.$$
This formula is essential for computing $h'(0)$.
3. **Apply the quotient rule at $x=0$:**
$$h'(0) = \frac{g(0)f'(0) - f(0)g'(0)}{[g(0)]^2}.$$
4. **Substitute the given values:**
- $g(0) = 5$
- $g'(0) = 6$
- $f(0) = 1$
- $f'(0) = 9$
So,
$$h'(0) = \frac{5 \times 9 - 1 \times 6}{5^2} = \frac{45 - 6}{25} = \frac{39}{25}.$$
5. **Identify the correct expression:** The expression that matches the quotient rule formula is option A:
$$\frac{g(0)f'(0) - f(0)g'(0)}{[g(0)]^2}.$$
**Final answer:** Option A is the correct expression to compute $h'(0)$.