Given that f and g are differentiable functions, which of the following correctly expresses the derivative of $v\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ in terms of $f$, $g$, $f^{\prime }$, and $g^{\prime }$?

• A. $v^{\prime }\left(x\right)=\frac{f^{\prime }\left(x\right)g\left(x\right)-f\left(x\right)g^{\prime }\left(x\right)}{{g\left(x\right)}^2}$
• B. $v^{\prime }\left(x\right)=\frac{f^{\prime }\left(x\right)g\left(x\right)+f\left(x\right)g^{\prime }\left(x\right)}{{g\left(x\right)}^2}$
• C. $v^{\prime }\left(x\right)=f^{\prime }\left(x\right)g^{\prime }\left(x\right)$
• D. $v^{\prime }\left(x\right)=\frac{f^{\prime }\left(x\right)g^{\prime }\left(x\right)}{}$

Answer: (A)

Verified step by step guidance

1. Step 1: Recognize that the function v(x) = f(x)/g(x) is a quotient of two differentiable functions. To find its derivative, we need to apply the Quotient Rule.
2. Step 2: Recall the Quotient Rule formula: If v(x) = f(x)/g(x), then v'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2. This rule is derived from the product rule and the chain rule.
3. Step 3: Identify the components of the formula: f(x) is the numerator, g(x) is the denominator, f'(x) is the derivative of the numerator, and g'(x) is the derivative of the denominator.
4. Step 4: Substitute the components into the Quotient Rule formula. The numerator of the derivative will be f'(x)g(x) - f(x)g'(x), and the denominator will be (g(x))^2.
5. Step 5: Compare the given options to the derived formula. The correct expression for v'(x) is (f'(x)g(x) - f(x)g'(x)) / (g(x))^2, as it matches the Quotient Rule.