Question

Let \(h(x)=f(g(x))\) and \(k(x)=g(g(x))\) Use the table to compute the following derivatives. a. \(h^{\prime}(1)\) b. \(h^{\prime}(2)\) c. \(h^{\prime}(3)\) d. \(k^{\prime}(3)\) e. \(k^{\prime}(1)\) f. \(k^{\prime}(5)\) $$\begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline f^{\prime}(x) & -6 & -3 & 8 & 7 & 2 \\ g(x) & 4 & 1 & 5 & 2 & 3 \\ g^{\prime}(x) & 9 & 7 & 3 & -1 & -5 \end{array}$$

Short answer

Question: Use the given table to find the requested values of the derivatives \(h'(x)\) and \(k'(x)\) for the specified x-values. | x | f'(x) | g(x) | g'(x) | |-----|-------|------|-------| | 1 | 7 | 4 | 9 | | 2 | -6 | 1 | 7 | | 3 | 2 | 5 | 3 | | 4 | 1 | 3 | -1 | | 5 | 4 | 2 | -5 | Given functions: \(h(x) = f(g(x))\) and \(k(x) = g(g(x))\) Find: a) \(h'(1)\) b) \(h'(2)\) c) \(h'(3)\) d) \(k'(3)\) e) \(k'(1)\) f) \(k'(5)\) Answer: a) \(h'(1) = 63\) b) \(h'(2) = -42\) c) \(h'(3) = 6\) d) \(k'(3) = -15\) e) \(k'(1) = -9\) f) \(k'(5) = -15\)

Step-by-step solution

Find f'(g(1)) and g'(1) using the table

For x=1, we have g(1) = 4 and g'(1) = 9. Now, we need to find f'(g(1)) = f'(4) = 7 (using the table).

Compute h'(1) using the Chain Rule

We have h'(x) = f'(g(x)) * g'(x), so h'(1) = f'(4) * g'(1) = 7 * 9 = 63. #b. h'(2):#

Find f'(g(2)) and g'(2) using the table

For x=2, we have g(2) = 1 and g'(2) = 7. Now, we need to find f'(g(2)) = f'(1) = -6 (using the table).

Compute h'(2) using the Chain Rule

We have h'(x) = f'(g(x)) * g'(x), so h'(2) = f'(1) * g'(2) = -6 * 7 = -42. #c. h'(3):#

Find f'(g(3)) and g'(3) using the table

For x=3, we have g(3) = 5 and g'(3) = 3. Now, we need to find f'(g(3)) = f'(5) = 2 (using the table).

Compute h'(3) using the Chain Rule

We have h'(x) = f'(g(x)) * g'(x), so h'(3) = f'(5) * g'(3) = 2 * 3 = 6. #d. k'(3): #

Find g'(g(3)) and g'(3) using the table

For x=3, we have g(3) = 5 and g'(3) = 3. Now, we need to find g'(g(3)) = g'(5) = -5 (using the table).

Compute k'(3) using the Chain Rule

We have k'(x) = g'(g(x)) * g'(x), so k'(3) = g'(5) * g'(3) = -5 * 3 = -15. #e. k'(1): #

Find g'(g(1)) and g'(1) using the table

For x=1, we have g(1) = 4 and g'(1) = 9. Now, we need to find g'(g(1)) = g'(4) = -1 (using the table).

Compute k'(1) using the Chain Rule

We have k'(x) = g'(g(x)) * g'(x), so k'(1) = g'(4) * g'(1) = -1 * 9 = -9. #f. k'(5):#

Find g'(g(5)) and g'(5) using the table

For x=5, we have g(5) = 3 and g'(5) = -5. Now, we need to find g'(g(5)) = g'(3) = 3 (using the table).

Compute k'(5) using the Chain Rule

We have k'(x) = g'(g(x)) * g'(x), so k'(5) = g'(3) * g'(5) = 3 * (-5) = -15.

Key concepts

Understanding Derivatives

Derivatives play a crucial role in calculus, helping us measure how a function changes as its input changes. This forms the basis of differential calculus. When we talk about the derivative of a function, we are essentially looking at the rate of change or the slope of the function at any given point.
For example, if you think of a car moving, the derivative would give you the speed of the car at any specific moment. Quite simply, it tells us how fast or slow something is changing.

### Calculating Derivatives

• The derivative of a function \( f(x) \) is often denoted by \( f'(x) \) or \( \frac{df}{dx} \).
• For simple functions like \( x^2 \), the derivative is \( 2x \), meaning the slope increases as \( x \) increases.
• Using rules like the power rule, product rule, and chain rule helps in calculating derivatives of more complex functions.

In our particular exercise, we are using the chain rule to find the derivative of composite functions. Each derivative relies on values from the table to calculate specific rates of change at given points.

Exploring Composite Functions

A composite function is essentially a combination of two or more functions. Imagine you have one function inside another, like a Russian doll. This layering is where the term "composite" comes in.
For example, if you have two functions, \( f(x) \) and \( g(x) \), a composite function \( h(x) \) can be expressed as \( h(x) = f(g(x)) \). This means first apply \( g(x) \), and then take that result and apply \( f \) to it.

### How Composite Functions Work

• They involve performing one function after another, leading to a new result.
• This requires careful attention to detail when computing values or derivatives.
• In terms of derivatives, the chain rule often applies because of this function nesting.

In the exercise, we have two main composite functions \( h(x) = f(g(x)) \) and \( k(x) = g(g(x)) \). Using values like \( g(1), g(2), \) and their derivatives, we can unravel these layered functions to calculate desired derivatives at specific points.

The Importance of Understanding in Mathematics Education

Mathematics education aims to provide students not just with solutions, but comprehension. Understanding concepts like the chain rule helps students apply mathematical principles to solve real-world problems efficiently.

### Learning Strategies

• First, familiarize yourself with basic functions and derivatives before tackling composites.
• Use visual aids and graphs to see how functions work and interact.
• Practice problems help cement understanding and increase confidence.

When students grasp ideas like the chain rule, they achieve a deeper understanding of calculus, beyond memorizing formulas. Through exercises like the provided one, students learn to navigate through complex problems, enhancing their overall problem-solving skills.