f(t)=(〖4t〗^2-t)(t^3-〖8t〗^2+12) derivatives

To find the derivative of the function \( f(t) = (4t^2 - t)(t^3 - 8t^2 + 12) \), we will use the **product rule** for differentiation. The product rule states that:

\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\]

Here, let:
- \( u(t) = 4t^2 - t \)
- \( v(t) = t^3 - 8t^2 + 12 \)

### Step 1: Differentiate \( u(t) \) and \( v(t) \)
We first need to compute the derivatives of \( u(t) \) and \( v(t) \):

\[
u'(t) = \frac{d}{dt} (4t^2 - t) = 8t - 1
\]

\[
v'(t) = \frac{d}{dt} (t^3 - 8t^2 + 12) = 3t^2 - 16t
\]

### Step 2: Apply the product rule
Now apply the product rule:

\[
f'(t) = u'(t) \cdot v(t) + u(t) \cdot v'(t)
\]

Substitute \( u'(t) \), \( u(t) \), \( v'(t) \), and \( v(t) \) into this equation:

\[
f'(t) = (8t - 1)(t^3 - 8t^2 + 12) + (4t^2 - t)(3t^2 - 16t)
\]

### Step 3: Simplify each part

#### Part 1: Simplify \( (8t - 1)(t^3 - 8t^2 + 12) \)

\[
(8t - 1)(t^3 - 8t^2 + 12) = 8t(t^3 - 8t^2 + 12) - (t^3 - 8t^2 + 12)
\]

\[
= 8t^4 - 64t^3 + 96t - (t^3 - 8t^2 + 12)
\]

\[
= 8t^4 - 64t^3 + 96t - t^3 + 8t^2 - 12
\]

\[
= 8t^4 - 65t^3 + 8t^2 + 96t - 12
\]

#### Part 2: Simplify \( (4t^2 - t)(3t^2 - 16t) \)

\[
(4t^2 - t)(3t^2 - 16t) = 4t^2(3t^2 - 16t) - t(3t^2 - 16t)
\]

\[
= 12t^4 - 64t^3 - 3t^3 + 16t^2
\]

\[
= 12t^4 - 67t^3 + 16t^2
\]

### Step 4: Combine the two parts

Now combine the results from Part 1 and Part 2:

\[
f'(t) = (8t^4 - 65t^3 + 8t^2 + 96t - 12) + (12t^4 - 67t^3 + 16t^2)
\]

\[
= 8t^4 + 12t^4 - 65t^3 - 67t^3 + 8t^2 + 16t^2 + 96t - 12
\]

\[
= 20t^4 - 132t^3 + 24t^2 + 96t - 12
\]

Thus, the derivative of \( f(t) \) is:

\[
f'(t) = 20t^4 - 132t^3 + 24t^2 + 96t - 12
\]

---

Would you like a breakdown of any step in detail? Here are some related questions:

1. What is the product rule for differentiation, and when do we use it?
2. Can you explain how to find the derivative of a polynomial function?
3. What is the power rule in differentiation, and how is it applied here?
4. How would you apply the chain rule if a composite function were involved?
5. What strategies can simplify the algebraic expansion process?

**Tip:** Always simplify polynomial terms before applying derivative rules to make the process more efficient.

Learn how to find the derivative of the function f(t) = (4t^2 - t)(t^3 - 8t^2 + 12) using the product rule. This guide covers key calculus concepts, including differentiation and simplification techniques, making it ideal for high school or college students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation