Question

Assume both the graphs of f and g pass through the point \((3,2), f^{\prime}(3)=5,\) and \(g^{\prime}(3)=-10 .\) If \(p(x)=f(x) g(x)\) and \(q(x)=f(x) / g(x),\) find the following derivatives. $$p^{\prime}(3)$$

Short answer

#tag_title# Step 4: Apply the quotient rule to find q'(3) #tag_content# Given, q(x) = f(x)/g(x) and f'(3) = 5 and g'(3) = -10. We need to find q'(3). Since q'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2, we can plug in the known values for x = 3 to obtain q'(3). First, recall that both f(3) = g(3) = 2. Now, let's substitute these values into the equation of q'(x): q'(3) = (f'(3)g(3) - f(3)g'(3)) / (g(3))^2 q'(3) = ((5)(2) - (2)(-10)) / (2)^2 #tag_title# Step 5: Calculate q'(3) #tag_content# Now, we can calculate q'(3) using the values obtained in step 4: q'(3) = (10 + 20) / 4 q'(3) = 30 / 4 q'(3) = 7.5 Thus, the derivative of the function q(x) = f(x)/g(x) at x=3 is q'(3) = 7.5. #Short Answer# At x = 3, the derivative of the product function p(x) = f(x)g(x) is p'(3) = -10, and the derivative of the quotient function q(x) = f(x)/g(x) is q'(3) = 7.5.

Step-by-step solution

Identify the product and quotient rules

To find the derivatives of p(x) and q(x), we need to use the product and quotient rules. The product rule states that if p(x) = f(x)g(x), then p'(x) = f'(x)g(x) + f(x)g'(x). The quotient rule states that if q(x) = f(x)/g(x), then q'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2. Now, let's apply these rules to find p'(3).

Apply the product rule to find p'(3)

Given, p(x) = f(x)g(x) and f'(3) = 5 and g'(3) = -10. We need to find p'(3). Since p'(x) = f'(x)g(x) + f(x)g'(x), we can plug in the known values for x = 3 to obtain p'(3). First, we know that both f and g pass through the point (3, 2). Therefore, f(3) = g(3) = 2. Now, let's substitute these values into the equation of p'(x): p'(3) = f'(3)g(3) + f(3)g'(3) p'(3) = (5)(2) + (2)(-10)

Calculate p'(3)

Now, we can calculate p'(3) using the values obtained in step 2: p'(3) = 10 - 20 p'(3) = -10 Thus, the derivative of the function p(x) = f(x)g(x) at x=3 is p'(3) = -10.

Key concepts

Product Rule

Taking derivatives of products is made easier by using the product rule. This rule helps us find the derivative of a function that is the product of two other functions. For a function defined as \(p(x) = f(x)g(x)\), the product rule states that the derivative \(p'(x)\) is given by:\[p'(x) = f'(x)g(x) + f(x)g'(x)\]Here's how it works:

• Differentiate the first function and multiply it by the second function as it is.
• Add the first function, unchanged, multiplied by the derivative of the second function.

Understanding this concept helps in analyzing the variations of functions that are products. In our example, since the values at \(x = 3\) are provided, we simply plug them into the formula. This step-by-step substitution leads to finding \(p'(3) = -10\).

Quotient Rule

The quotient rule is a derivative rule designed to handle fractions of functions. If you have a function \(q(x)\) defined as the ratio of two functions, \(q(x) = \frac{f(x)}{g(x)}\), then the derivative \(q'(x)\) is calculated by:\[q'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}\]To apply this rule:

• Differentiate the numerator \(f(x)\) and multiply by the denominator \(g(x)\), as is.
• Subtract the product of the numerator \(f(x)\), unchanged, and the derivative of the denominator \(g'(x)\).
• Divide the entire expression by the square of the denominator, \((g(x))^2\).

This method ensures precision, particularly when dealing with functions not easily simplified into simpler expressions.

Derivative Calculation

Derivative calculation is a fundamental aspect of calculus, showing how a function changes at any point. The derivative itself represents the rate of change or the slope of the function at a specific point. In practical terms:

• Derivatives help in understanding how values fluctuate when inputs change slightly.
• They are used to find minimum and maximum values (optimizing functions) and to solve real-world problems related to motion, growth, and areas.

The process involves:

• Identifying which rules or combinations best fit the specific function.
• Applying those rules accurately to calculate or simplify derivative functions.

Using the product and quotient rules, one can efficiently determine derivative expressions and evaluate them at particular points, much like finding \(p'(3)\) using the provided function values.