Final answer:
The derivative of the function f(x) = -5x^4 + 7^{-2x^2} is found by applying the power rule to the first term, resulting in -20x^3, and the chain rule to the second term, resulting in -4x*ln(7)*7^{-2x^2}.
Step-by-step explanation:
To find the derivative of the function f(x) = -5x^4 + 7^{-2x^2}, we use the power rule and the chain rule for differentiation. For the first term, -5x^4, the power rule states that the derivative of x^n is n*x^(n-1), so the derivative is -20x^3. For the second term, 7^{-2x^2}, we apply the chain rule: the derivative of a^u, where u is a function of x, is (ln(a)*a^u)*(du/dx). Since u = -2x^2, du/dx is -4x, and the base, a, is 7, the derivative of the second term is -4x*ln(7)*7^{-2x^2}.
Combining these, we get the derivative:
f'(x) = -20x^3 - 4x*ln(7)*7^{-2x^2}.