https://file.aiquickdraw.com/mathbot/file/17290359238411dkcito0.jpg

The problem asks to **find the derivative of the function**:

\[
g(t) = 6 \cdot \left( \sin(t) \right)^8
\]

Let's solve this step-by-step using the **product rule** and **chain rule**.

### Step 1: Rewrite the function
\[
g(t) = 6 \cdot u(t), \quad \text{where} \quad u(t) = \left( \sin(t) \right)^8
\]

Since the constant \(6\) is just a multiplier, we can focus on finding the derivative of \(u(t)\) and multiply the result by 6.

---

### Step 2: Differentiate \(u(t) = (\sin(t))^8\) using the chain rule

The **chain rule** states:
\[
\frac{d}{dt} \left( f(g(t)) \right) = f'(g(t)) \cdot g'(t)
\]

In this case, \(f(x) = x^8\) and \(g(t) = \sin(t)\). First, differentiate the outer function:

\[
\frac{d}{dt} \left( \sin(t) \right)^8 = 8 \cdot \left( \sin(t) \right)^7 \cdot \frac{d}{dt} \left( \sin(t) \right)
\]

Now, differentiate the inner function \(\sin(t)\):
\[
\frac{d}{dt} \left( \sin(t) \right) = \cos(t)
\]

Thus:
\[
\frac{d}{dt} \left( \sin(t) \right)^8 = 8 \cdot \left( \sin(t) \right)^7 \cdot \cos(t)
\]

---

### Step 3: Multiply by the constant \(6\)
The derivative of \(g(t) = 6 \cdot \left( \sin(t) \right)^8\) is:
\[
g'(t) = 6 \cdot \left( 8 \cdot \left( \sin(t) \right)^7 \cdot \cos(t) \right)
\]

Simplify:
\[
g'(t) = 48 \cdot \left( \sin(t) \right)^7 \cdot \cos(t)
\]

---

### Final Answer:
\[
g'(t) = 48 \cdot \left( \sin(t) \right)^7 \cdot \cos(t)
\]

---

Would you like further details on the chain rule or a similar problem to practice?

---

### 5 Related Questions:
1. How does the chain rule work in multivariable calculus?
2. What is the derivative of \(\cos^4(t)\)?
3. How do you apply the product rule to two functions?
4. What are common mistakes when differentiating trigonometric functions?
5. How do you find second derivatives of functions like \( \sin(t)^n \)?

### Tip:
When working with trigonometric functions raised to a power, always remember to apply the chain rule carefully, accounting for both the outer power and the inner trigonometric derivative.

Find the derivative of the function g(t) = 6 * (sin(t))^8.

Find the derivative of the function g(t) = 6 * (sin(t))^8. This problem applies the chain rule to differentiate trigonometric functions raised to a power, providing a step-by-step solution for advanced 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