Suppose that
f(5) = 1, f '(5) = 7, g(5) = −4, and g'(5) = 5.
Find the following values.
(a)
(fg)'(5)

(b)

f
g

'(5)

(c)

g
f

'(5)

To solve for the derivative of the product and quotient of the functions \( f \) and \( g \) at \( x = 5 \), we will use the following differentiation rules:

1. **Product Rule** for \( (fg)' \):  
   \[
   (fg)' = f'g + fg'
   \]

2. **Quotient Rule** for \( \left(\frac{f}{g}\right)' \):  
   \[
   \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}
   \]

Now let's solve each part:

### (a) \( (fg)'(5) \)

Using the **Product Rule**:
\[
(fg)'(5) = f'(5)g(5) + f(5)g'(5)
\]
Substitute the given values:
\[
(fg)'(5) = 7(-4) + 1(5) = -28 + 5 = -23
\]

Thus, \( (fg)'(5) = -23 \).

---

### (b) \( \left(\frac{f}{g}\right)'(5) \)

Using the **Quotient Rule**:
\[
\left(\frac{f}{g}\right)'(5) = \frac{f'(5)g(5) - f(5)g'(5)}{g(5)^2}
\]
Substitute the given values:
\[
\left(\frac{f}{g}\right)'(5) = \frac{7(-4) - 1(5)}{(-4)^2} = \frac{-28 - 5}{16} = \frac{-33}{16}
\]

Thus, \( \left(\frac{f}{g}\right)'(5) = \frac{-33}{16} \).

---

### (c) \( \left(\frac{g}{f}\right)'(5) \)

Using the **Quotient Rule** again for \( \left(\frac{g}{f}\right)' \):
\[
\left(\frac{g}{f}\right)'(5) = \frac{g'(5)f(5) - g(5)f'(5)}{f(5)^2}
\]
Substitute the given values:
\[
\left(\frac{g}{f}\right)'(5) = \frac{5(1) - (-4)(7)}{(1)^2} = \frac{5 + 28}{1} = 33
\]

Thus, \( \left(\frac{g}{f}\right)'(5) = 33 \).

---

### Final Answers:

(a) \( (fg)'(5) = -23 \)

(b) \( \left(\frac{f}{g}\right)'(5) = \frac{-33}{16} \)

(c) \( \left(\frac{g}{f}\right)'(5) = 33 \)

Let me know if you want further details on any step. Here are some follow-up questions to practice related concepts:

1. What is the product rule for three functions?
2. How do you derive the quotient rule?
3. What happens to the quotient rule if \( g(5) = 0 \)?
4. Can you apply the chain rule with these functions? If so, how?
5. How would these derivatives change if we had higher-order derivatives?

**Tip:** The product rule is useful even when dealing with more complex combinations of functions, like compositions or multi-variable cases.

Suppose that f(5) = 1, f '(5) = 7, g(5) = −4, and g'(5) = 5. Find the following values. (a) (fg)'(5), (b) (f/g)'(5), (c) (g/f)'(5)

Learn how to solve for derivatives using the product and quotient rules with given function values and derivatives at a specific point. This problem covers key calculus concepts and differentiation techniques. Suitable for advanced high school or college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation