Derivatives from a table Use the following table to find the given derivatives. <IMAGE>
d/dx (xf(x) / g(x)) |x=4

Verified step by step guidance

1. Step 1: Identify the function to differentiate. We have the function \( \frac{x f(x)}{g(x)} \).
2. Step 2: Apply the quotient rule for derivatives. The quotient rule states that if you have a function \( \frac{u(x)}{v(x)} \), its derivative is \( \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). Here, \( u(x) = x f(x) \) and \( v(x) = g(x) \).
3. Step 3: Differentiate \( u(x) = x f(x) \) using the product rule. The product rule states that \( (uv)' = u'v + uv' \). So, \( u'(x) = 1 \cdot f(x) + x \cdot f'(x) \).
4. Step 4: Differentiate \( v(x) = g(x) \). The derivative is \( v'(x) = g'(x) \).
5. Step 5: Substitute \( u'(x) \), \( v(x) \), \( u(x) \), and \( v'(x) \) into the quotient rule formula and evaluate at \( x = 4 \).

Key Concepts

Product Rule

The Product Rule is a fundamental principle in calculus used to differentiate products of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential for finding the derivative of the function in the question, which involves the product of x and f(x).

Quotient Rule

The Quotient Rule is another key differentiation rule used when dealing with the division of two functions. If you have a function h(x) = u(x)/v(x), the derivative is given by (u'v - uv')/v^2. This rule is crucial for differentiating the expression xf(x)/g(x) in the question, as it involves both a product and a quotient of functions.

Evaluating Derivatives at a Point

Evaluating derivatives at a specific point involves substituting the value of x into the derivative function after it has been calculated. In this case, after applying the Product and Quotient Rules, you will substitute x = 4 into the resulting derivative expression to find the specific value of the derivative at that point. This step is essential for obtaining the final answer.