Find the derivative of the function.
​m(t)equalsnegative 6 t left parenthesis 5 t Superscript 8 Baseline minus 1 right parenthesis Superscript 5
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Part 1
m prime left parenthesis t right parenthesisequals
  
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We are tasked with finding the derivative of the function:  
\[ m(t) = -6t(5t^8 - 1)^5 \]

This is a product of two functions:  
- \( u(t) = -6t \)
- \( v(t) = (5t^8 - 1)^5 \)

We'll use the **product rule** for differentiation:
\[
\frac{d}{dt}(u(t)v(t)) = u'(t)v(t) + u(t)v'(t)
\]

### Step 1: Differentiate \( u(t) = -6t \)
\[
u'(t) = -6
\]

### Step 2: Differentiate \( v(t) = (5t^8 - 1)^5 \) using the chain rule.
Let \( g(t) = 5t^8 - 1 \), then \( v(t) = g(t)^5 \). Applying the chain rule:
\[
v'(t) = 5 \cdot g(t)^4 \cdot g'(t)
\]
Now, calculate \( g'(t) \):
\[
g'(t) = \frac{d}{dt}(5t^8 - 1) = 40t^7
\]
Thus,
\[
v'(t) = 5(5t^8 - 1)^4 \cdot 40t^7 = 200t^7(5t^8 - 1)^4
\]

### Step 3: Apply the product rule
Using the product rule formula:
\[
m'(t) = u'(t)v(t) + u(t)v'(t)
\]
Substitute \( u'(t) = -6 \), \( u(t) = -6t \), \( v(t) = (5t^8 - 1)^5 \), and \( v'(t) = 200t^7(5t^8 - 1)^4 \):
\[
m'(t) = (-6)(5t^8 - 1)^5 + (-6t)(200t^7(5t^8 - 1)^4)
\]
Simplify each term:
\[
m'(t) = -6(5t^8 - 1)^5 - 1200t^8(5t^8 - 1)^4
\]

This is the derivative of the given function.

Let me know if you'd like further clarification or a more detailed explanation.

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Here are 5 related questions you might consider:
1. How does the product rule for derivatives apply to other types of functions?
2. What is the chain rule, and how is it used in differentiation?
3. How do you simplify expressions after applying the product and chain rules?
4. What happens if you differentiate a product of three functions?
5. How do derivatives of higher powers of polynomials behave?

**Tip:** When using the chain rule, always remember to differentiate both the outer and inner functions.

Find the derivative of the function m(t) = -6t(5t^8 - 1)^5.

Learn how to find the derivative of the function m(t) = -6t(5t^8 - 1)^5 by applying both the product rule and the chain rule. This step-by-step guide explains the key calculus concepts and provides detailed explanations. Suitable for college-level calculus students.

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