A graph of a the piecewise linear function 𝑓 (𝑥) and table of the functions 𝑔(𝑥) and
𝑔
′
(𝑥) are shown below.
1
(a) Given ℎ(𝑥) = 𝑓 (𝑥) + 𝑔(𝑥), find ℎ
′
(1)
(b) Given 𝑘(𝑥) = 𝑓 (𝑥)𝑔(𝑥), find 𝑘
′
(1).
(c) Given 𝑙(𝑥) =
𝑓 (𝑥)
𝑔(𝑥)
, find 𝑙
′
(3).
(d) Given 𝑚(𝑥) =
𝑔(𝑥)
√
𝑥
, find 𝑚′
(4)

### Problem Breakdown:

We are given several functions and their derivatives. The tasks require applying standard rules for differentiation, such as the sum rule, product rule, quotient rule, and chain rule. Let's break each part down.

#### (a) Given \( h(x) = f(x) + g(x) \), find \( h'(1) \)

**Sum Rule**:
\[
h'(x) = f'(x) + g'(x)
\]
Therefore, \( h'(1) = f'(1) + g'(1) \).

To calculate this, we need to know \( f'(1) \) and \( g'(1) \). According to the table or graph, extract the values of \( f'(1) \) from the piecewise function and \( g'(1) \) from the given table.

---

#### (b) Given \( k(x) = f(x)g(x) \), find \( k'(1) \)

**Product Rule**:
\[
k'(x) = f'(x)g(x) + f(x)g'(x)
\]
So, \( k'(1) = f'(1)g(1) + f(1)g'(1) \).

Again, use the values of \( f'(1), f(1), g(1), g'(1) \) from the graph and table.

---

#### (c) Given \( l(x) = \frac{f(x)}{g(x)} \), find \( l'(3) \)

**Quotient Rule**:
\[
l'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
\]
Thus, \( l'(3) = \frac{f'(3)g(3) - f(3)g'(3)}{(g(3))^2} \).

Use the values of \( f'(3), f(3), g(3), g'(3) \) from the provided information.

---

#### (d) Given \( m(x) = \frac{g(x)}{\sqrt{x}} \), find \( m'(4) \)

This is a quotient of \( g(x) \) and \( \sqrt{x} \), so we will apply the **quotient rule** with the chain rule for the square root function.

For \( \sqrt{x} \), the derivative is:
\[
\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}
\]

Applying the **quotient rule**:
\[
m'(x) = \frac{g'(x)\sqrt{x} - g(x) \cdot \frac{1}{2\sqrt{x}}}{x}
\]
Substitute \( x = 4 \):
\[
m'(4) = \frac{g'(4)\sqrt{4} - g(4) \cdot \frac{1}{2\sqrt{4}}}{4}
\]

Now, use the values of \( g'(4) \) and \( g(4) \).

---

### Summary of Required Steps:

1. Find the values of \( f'(x), f(x), g'(x), g(x) \) at \( x = 1, 3, 4 \).
2. Apply the sum, product, and quotient rules to each problem.
3. Compute the derivatives.

Would you like to provide the table or graph for the function values, or proceed with further calculations?

A graph of the piecewise linear function f(x) and table of the functions g(x) and g'(x) are shown below. (a) Given h(x) = f(x) + g(x), find h'(1). (b) Given k(x) = f(x)g(x), find k'(1). (c) Given l(x) = f(x)/g(x), find l'(3). (d) Given m(x) = g(x)/√x, find m'(4).

Learn how to apply differentiation rules such as the sum rule, product rule, quotient rule, and chain rule to find the derivatives of composite functions like h(x) = f(x) + g(x), k(x) = f(x)g(x), and more. This step-by-step solution uses values from a piecewise linear graph and a function table, ideal for Grades 11-12 students studying calculus.

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