Answer :

The 5th derivative of the function f(x)= 19x5 + 11x3 + 11x2 is a constant, 2280. This is calculated by taking the derivative of the function five times.

In calculus, we take derivatives of functions to understand how they change. The 5th derivative of the function f(x)= 19x5 + 11x3 + 11x2, is the result of taking the derivative 5 times.

First derivative: f'(x)= 95x4 + 33x2 + 22x

Second derivative:= 380x3 + 66x + 22

Third derivative: = 1140x2 + 66

Fourth derivative: = 2280x + 0

Fifth derivative: = 2280

After the 5th derivative, the function f(x) becomes a constant and does not change any further, hence we stop at the 5th derivative.

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Answer:

2280

Step-by-step explanation:

f(x) = 19x⁵ + 11x³+ 11x²

1st derivative: (19×5)x^(5-1) + (11×3)x^(3-1) + (11×2)x^(2-1) = 95x⁴ + 33x² + 22x

2nd derivative: (95×4)x^(4-1) + (33×2)x^(2-1) + (22×1)x^(1-1) = 380x³ + 66x + 22

3rd derivative: (380×3)x^(3-1) + (66×1)x^(1-1) + 0 = 1140x² + 66

4th derivative: (1140×2)x^(2-1) + 0 = 2280x

5th derivative: (2280×1)x^(1-1) = 2280