f(x) = (2t^5 - t)(t^3 - 2t + 1) find f'(x)

To find the derivative of the function \( f(x) = (2t^5 - t)(t^3 - 2t + 1) \), we can use the product rule. The product rule states that if \( f(x) = g(x)h(x) \), then

\[
f'(x) = g'(x)h(x) + g(x)h'(x)
\]

Let's define:
- \( g(t) = 2t^5 - t \)
- \( h(t) = t^3 - 2t + 1 \)

### Step 1: Find \( g'(t) \)
The derivative of \( g(t) = 2t^5 - t \) is:

\[
g'(t) = 10t^4 - 1
\]

### Step 2: Find \( h'(t) \)
The derivative of \( h(t) = t^3 - 2t + 1 \) is:

\[
h'(t) = 3t^2 - 2
\]

### Step 3: Apply the product rule
Now, using the product rule:

\[
f'(x) = g'(t)h(t) + g(t)h'(t)
\]

Substitute the expressions for \( g(t) \), \( g'(t) \), \( h(t) \), and \( h'(t) \):

\[
f'(x) = (10t^4 - 1)(t^3 - 2t + 1) + (2t^5 - t)(3t^2 - 2)
\]

### Step 4: Expand the expression
Let's expand both terms:

1. Expand \( (10t^4 - 1)(t^3 - 2t + 1) \):

\[
(10t^4)(t^3) + (10t^4)(-2t) + (10t^4)(1) - (t^3) + (2t) - 1
= 10t^7 - 20t^5 + 10t^4 - t^3 + 2t - 1
\]

2. Expand \( (2t^5 - t)(3t^2 - 2) \):

\[
(2t^5)(3t^2) + (2t^5)(-2) - (t)(3t^2) + (t)(2)
= 6t^7 - 4t^5 - 3t^3 + 2t
\]

### Step 5: Combine like terms
Now, add the expanded terms together:

\[
f'(x) = (10t^7 - 20t^5 + 10t^4 - t^3 + 2t - 1) + (6t^7 - 4t^5 - 3t^3 + 2t)
\]

Combine like terms:

\[
f'(x) = 16t^7 - 24t^5 + 10t^4 - 4t^3 + 4t - 1
\]

So, the derivative of the function is:

\[
f'(x) = 16t^7 - 24t^5 + 10t^4 - 4t^3 + 4t - 1
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you apply the chain rule in differentiation?
2. What is the derivative of a polynomial function?
3. How do you differentiate a product of three or more functions?
4. How can you use the quotient rule in differentiation?
5. What are some applications of derivatives in real-world problems?

### Tip:
When differentiating, always start by identifying whether you need to apply the product rule, quotient rule, or chain rule to simplify the process.

Learn how to find the derivative of the function f(x) = (2t^5 - t)(t^3 - 2t + 1) using the product rule. Detailed step-by-step solution including differentiation of polynomial functions. Suitable for advanced high school students studying calculus.

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