To find \(\frac{dy}{dx}\), we will use the product rule and chain rule. The given function is:

\[
y = \frac{6 \cos(5x)}{x^5}
\]

### Step 1: Rewrite the function
We can rewrite \(y\) as:

\[
y = 6 \cos(5x) \cdot x^{-5}
\]

Now we will differentiate using the **product rule**, which is:

\[
\frac{d}{dx}[u \cdot v] = u'v + uv'
\]

Here, let:

- \( u = 6 \cos(5x) \)
- \( v = x^{-5} \)

### Step 2: Differentiate \(u\) and \(v\)
- The derivative of \(u = 6 \cos(5x)\) with respect to \(x\):
  \[
  u' = -6 \cdot \sin(5x) \cdot 5 = -30 \sin(5x)
  \]

- The derivative of \(v = x^{-5}\) with respect to \(x\):
  \[
  v' = -5x^{-6}
  \]

### Step 3: Apply the product rule
Now, apply the product rule to find \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = u'v + uv'
\]

Substitute the values of \(u\), \(v\), \(u'\), and \(v'\):

\[
\frac{dy}{dx} = (-30 \sin(5x)) \cdot x^{-5} + (6 \cos(5x)) \cdot (-5x^{-6})
\]

### Step 4: Simplify the expression
Now, simplify each term:

\[
\frac{dy}{dx} = -30 \sin(5x) x^{-5} - 30 \cos(5x) x^{-6}
\]

Thus, the derivative is:

\[
\frac{dy}{dx} = -30 x^{-5} \sin(5x) - 30 x^{-6} \cos(5x)
\]

This is the final expression for \(\frac{dy}{dx}\).

Would you like further details on any of these steps?

---

Here are 5 related questions for you:
1. What is the product rule in differentiation?
2. How is the chain rule applied when differentiating trigonometric functions?
3. Can you apply implicit differentiation to find the derivative of a function?
4. How do you differentiate a function with a negative exponent?
5. What is the significance of simplifying a derivative?

**Tip:** Always simplify your functions as much as possible before differentiating; it often makes the process much easier!

This detailed solution explains how to differentiate the function y = 6cos(5x)/x^5 using the product rule and chain rule. The solution covers trigonometric functions and power rules in calculus, suitable for advanced high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation