91Ó°ÊÓ

When you have a function that is the product of two other functions, we use the product rule to differentiate it. The product rule formula states that for two functions, say \( u \) and \( v \), the derivative of their product is given by: \[ (u \times v)' = u'v + uv' \]. In simpler terms, it means we first differentiate the first function while keeping the second function constant, and then differentiate the second function while keeping the first function constant, and sum up these two results. In our exercise, we identified our two functions as: \( u = x^7 \) and \( v = (3x^4 + 12x - 1)^2 \).
This method allows us to systematically handle the differentiation process for each part of the function.

When dealing with composite functions, where one function is nested inside another, we use the chain rule for differentiation. The chain rule can be written as: \[ \frac{d}{dx} f(g(x)) = f'(g(x)) \times g'(x) \]. This means we first differentiate the outer function, leaving the inner function unchanged, then multiply by the derivative of the inner function.In our situation, the function \( v \) is a composite function where: \( v = (3x^4 + 12x - 1)^2 \). Let’s break it down:

• Set \( w = 3x^4 + 12x - 1 \)
• So \( v = w^2 \)

Applying the chain rule here:\[ v' = 2w \frac{dw}{dx}\].
Afterwards, we find the derivative of \( w \) with respect to \( x \):
\[ \frac{dw}{dx} = 12x^3 + 12 \].Combining these steps gives us:\[ v' = 2(3x^4 + 12x - 1)(12x^3 + 12) \].This whole process breaks down a potentially complicated differentiation task into manageable parts.

Once we have the derivatives of the individual components using the product and chain rules, we can combine them to find the derivative of the original function. Starting from our product rule formula: \( y = x^7 (3x^4 + 12x - 1)^2 \), we identified:
\[ u = x^7, u' = 7x^6 \], and \[ v = (3x^4 + 12x - 1)^2 , v' = 2(3x^4 + 12x - 1)(12x^3 + 12) \].Plugging these back into the product rule, we get:
\[ y' = x^7 \times 2(3x^4 + 12x - 1)(12x^3 + 12) + 7x^6 \times (3x^4 + 12x - 1)^2 \]. Finally, substitute back the expressions and simplify if necessary. By following these steps, you can accurately handle the differentiation tasks of various composite and product functions using the product and chain rules.