If
f
(
x
)
=
8
(
sin
(
x
)
)
x
, find
f
'
(
3
)
.

We are given the function \( f(x) = 8 \cdot \frac{\sin(x)}{x} \), and we are tasked with finding \( f'(3) \), the derivative of \( f(x) \) evaluated at \( x = 3 \).

### Step 1: Apply the Product Rule

We can express \( f(x) \) as:

\[
f(x) = 8 \cdot \left( \frac{\sin(x)}{x} \right)
\]

To differentiate this, we will apply the **product rule** and the **quotient rule**. The product rule states that for two functions \( u(x) \) and \( v(x) \), the derivative is:

\[
(uv)' = u'v + uv'
\]

Here, let:

- \( u(x) = 8 \)
- \( v(x) = \frac{\sin(x)}{x} \)

Since \( u(x) = 8 \), its derivative \( u'(x) = 0 \), so we only need to focus on differentiating \( v(x) = \frac{\sin(x)}{x} \).

### Step 2: Differentiate \( v(x) = \frac{\sin(x)}{x} \)

To differentiate \( v(x) = \frac{\sin(x)}{x} \), we use the **quotient rule**:

\[
\left( \frac{\sin(x)}{x} \right)' = \frac{x \cos(x) - \sin(x)}{x^2}
\]

### Step 3: Differentiate \( f(x) \)

Now, using the product rule:

\[
f'(x) = u'(x)v(x) + u(x)v'(x) = 0 + 8 \cdot \frac{x \cos(x) - \sin(x)}{x^2}
\]

So:

\[
f'(x) = 8 \cdot \frac{x \cos(x) - \sin(x)}{x^2}
\]

### Step 4: Evaluate at \( x = 3 \)

Now, we substitute \( x = 3 \) into the derivative:

\[
f'(3) = 8 \cdot \frac{3 \cos(3) - \sin(3)}{3^2} = 8 \cdot \frac{3 \cos(3) - \sin(3)}{9}
\]

\[
f'(3) = \frac{8}{9} \cdot (3 \cos(3) - \sin(3))
\]

### Step 5: Approximate the result

Using the approximate values:

- \( \cos(3) \approx -0.98999 \)
- \( \sin(3) \approx 0.14112 \)

Substitute these into the expression:

\[
f'(3) = \frac{8}{9} \cdot (3(-0.98999) - 0.14112)
\]

\[
f'(3) = \frac{8}{9} \cdot (-2.96997 - 0.14112) = \frac{8}{9} \cdot (-3.11109)
\]

\[
f'(3) \approx \frac{8}{9} \cdot (-3.11109) \approx -2.76408
\]

Thus, \( f'(3) \approx -2.76 \).

Let me know if you want a more detailed explanation or have any questions. Here are some related questions to further your understanding:

1. How do you apply the product rule in calculus?
2. What are some other examples of using the quotient rule?
3. How do trigonometric derivatives affect differentiation?
4. What is the significance of evaluating a derivative at a specific point?
5. Can you generalize this approach to other similar functions?

**Tip:** When differentiating a quotient, make sure you carefully apply the quotient rule, as it's a common source of mistakes!

Discover how to find the derivative of the function f(x) = 8(sin(x))/x and evaluate it at x = 3 using the product and quotient rules. This solution explores differentiation techniques and trigonometric derivatives, suitable for students in Grades 11-12.

Math Problem Statement

If f ( x )

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation