Final answer:
To find the derivative of the function f(x)=4e⁻²⁵ ⁻⁵¹⁰, apply the chain rule to compute the derivative of the exponent as an inner function and multiply it by the derivative of the outer function. The final derivative expression is found by substituting the inner function back into the derived equation.
Step-by-step explanation:
To find the derivative of the function f(x)=4e⁻²⁵ ⁻⁵⁵ⁱ⁰, we will apply the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
The function f(x) can be rewritten as f(x) = 4e-(2x5 + 5x10). Let g(x) = -(2x5 + 5x10), so f(x) = 4eg(x). Now we apply the chain rule to f(x).
f'(x) = 4eg(x) · g'(x).
To find g'(x) we differentiate g(x) with respect to x.
g'(x) = -((2·5)x4 + (5·10)x9)
g'(x) = -10x4 - 50x9.
Then we substitute g'(x) back into the derivative of f(x).
f'(x) = 4eg(x) · (-10x4 - 50x9)
f'(x) = -40x4eg(x) - 200x9eg(x)
Finally, we substitute back g(x) into the expression:
f'(x) = -40x4e-(2x5 + 5x10) - 200x9e-(2x5 + 5x10)