Answer :
The derivative of f(x) = 3e^(-4x^8 + 4x^9) is found by applying the chain rule: first, we take the derivative of the outer exponential function, which remains e^u, and then multiply it by the derivative of the inner polynomial, which is -32x^7 + 36x^8, resulting in f'(x) = 3e^(-4x^8 + 4x^9)(-32x^7 + 36x^8).
To use the chain rule to find the derivative of f(x) = 3e^(-4x^8 + 4x^9), we first identify the outer function and the inner function. The outer function is e^u where u is the inner function, which in this case is -4x^8 + 4x^9. The chain rule states that the derivative of a composed function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.
Step-by-step application:
Take the derivative of the outer function, which is e^u, giving us e^u.
Derive the inner function u = -4x^8 + 4x^9, which gives us du/dx = -32x^7 + 36x^8.
Multiply the derivative of the outer function by the derivative of the inner function: f'(x) = 3e^(-4x^8 + 4x^9)(-32x^7 + 36x^8).
This result represents the derivative of f(x) using the chain rule.
