Find the derivative of the function $z=\left(t e^{8 t}+e^{7 t}\right)^{9}$

Step 1 ：Given the function \(z=(t e^{8 t}+e^{7 t})^{9}\)

Step 2 ：We need to find its derivative. We can use the chain rule and the product rule to solve this problem.

Step 3 ：The chain rule is a formula to compute the derivative of a composite function. The outer function is \((...)^9\) and the inner function is \(t e^{8 t} + e^{7 t}\).

Step 4 ：The product rule is a formula used to find the derivative of a product of two or more functions, in this case \(t\) and \(e^{8 t}\).

Step 5 ：Applying the chain rule, we differentiate the outer function first, keeping the inner function as it is. This gives us \(9(t e^{8 t}+e^{7 t})^{8}\).

Step 6 ：Next, we differentiate the inner function using the product rule. The derivative of \(t e^{8 t}\) is \(e^{8 t} + 8 t e^{8 t}\) and the derivative of \(e^{7 t}\) is \(7 e^{7 t}\).

Step 7 ：Multiplying these together, we get \((t e^{8 t}+e^{7 t})^{8}(e^{8 t} + 8 t e^{8 t} + 7 e^{7 t})\).

Step 8 ：Simplifying this expression, we get \((t e^{8 t}+e^{7 t})^{8}(9 e^{8 t} + 72 t e^{8 t} + 63 e^{7 t})\).

Step 9 ：Final Answer: The derivative of the function \(z=(t e^{8 t}+e^{7 t})^{9}\) is \(\boxed{(t e^{8 t}+e^{7 t})^{8}(9 e^{8 t} + 72 t e^{8 t} + 63 e^{7 t})}\).