Hello!
To find the derivative of the function \(g(z) = 2z(8z^2 - 6z + 1)^4\), we can use the Generalized Power Rule, which states that the derivative of a function of the form \(u(x)^n\) is \(n \cdot u(x)^{n-1} \cdot u'(x)\), where \(u(x)\) is a differentiable function and \(n\) is a constant.
Let's apply the Generalized Power Rule step by step:
1. Find the derivative of \(2z\):
The derivative of \(2z\) with respect to \(z\) is \(2\).
2. Find the derivative of \(8z^2 - 6z + 1\) with respect to \(z\):
Using the Power Rule, the derivative of \(8z^2 - 6z + 1\) with respect to \(z\) is \(16z - 6\).
3. Apply the Generalized Power Rule to the function \(g(z) = 2z(8z^2 - 6z + 1)^4\):
\(g'(z) = 2 \cdot (8z^2 - 6z + 1)^4 \cdot (16z - 6) + 2z \cdot 4 \cdot (8z^2 - 6z + 1)^3\).
Simplify the expression:
\(g'(z) = 2 \cdot (8z^2 - 6z + 1)^4 \cdot (16z - 6) + 8z \cdot (8z^2 - 6z + 1)^3\).
So, the derivative of the function \(g(z) = 2z(8z^2 - 6z + 1)^4\) with respect to \(z\) is \(2 \cdot (8z^2 - 6z + 1)^4 \cdot (16z - 6) + 8z \cdot (8z^2 - 6z + 1)^3\).
Hope this help you!
If you have more questions, feel free to ask!