To find the derivative of the function $f(x) = (3x^2 - 4x + 5)^5 \cdot (4x^2 - 5x + 1)^7$, we can use the product rule and the chain rule.

Given:

$f(x) = u(x) \cdot v(x) \quad \text{where} \quad u(x) = (3x^2 - 4x + 5)^5 \quad \text{and} \quad v(x) = (4x^2 - 5x + 1)^7$

The derivative $f'(x)$ is given by the product rule:

$f'(x) = u'(x)v(x) + u(x)v'(x)$

Step 1: Differentiate $u(x) = (3x^2 - 4x + 5)^5$

Using the chain rule, let $g(x) = 3x^2 - 4x + 5$:

$u(x) = g(x)^5 \Rightarrow u'(x) = 5g(x)^4 \cdot g'(x)$

$g'(x) = 6x - 4 \Rightarrow u'(x) = 5(3x^2 - 4x + 5)^4 \cdot (6x - 4)$

Step 2: Differentiate $v(x) = (4x^2 - 5x + 1)^7$

Using the chain rule, let $h(x) = 4x^2 - 5x + 1$:

$v(x) = h(x)^7 \Rightarrow v'(x) = 7h(x)^6 \cdot h'(x)$

$h'(x) = 8x - 5 \Rightarrow v'(x) = 7(4x^2 - 5x + 1)^6 \cdot (8x - 5)$

Step 3: Combine using the product rule

$f'(x) = u'(x)v(x) + u(x)v'(x)$

$f'(x) = 5(3x^2 - 4x + 5)^4 \cdot (6x - 4) \cdot (4x^2 - 5x + 1)^7 + (3x^2 - 4x + 5)^5 \cdot 7(4x^2 - 5x + 1)^6 \cdot (8x - 5)$

This is the derivative of the given function $f(x)$.

Would you like a more detailed step-by-step explanation, or do you have any questions?

Related Questions

1. How do you apply the product rule when there are more than two functions?
2. Can you explain the chain rule in more detail with other examples?
3. How would you approach finding the second derivative of this function?
4. What happens if one of the factors of $f(x)$ is a trigonometric function?
5. How would you find the critical points of $f(x)$ using its derivative?

Tip

When using the product rule, carefully identify each component of the product and their derivatives separately before combining them. This helps to avoid mistakes, especially when working with complex functions.