Answer:
4th Order Derivative: 120
Step by sep solution:
To find the fourth-order derivative of the function f(x) = 5x^4, we can differentiate the third-order derivative f(3)(x) = d^3/dx^3 (5x^4) with respect to x:
f(3)(x) = d^3/dx^3 (5x^4) = 5 * d^3/dx^3 (x^4)
To find d^3/dx^3 (x^4), we differentiate the function x^4 three times:
d/dx (x^4) = 4x^3
d^2/dx^2 (x^4) = d/dx (4x^3) = 12x^2
d^3/dx^3 (x^4) = d/dx (12x^2) = 24x
Substituting this back into the expression for the third-order derivative, we get:
f(3)(x) = 5 * d^3/dx^3 (x^4) = 5 * 24x = 120x
Now we can differentiate f(3)(x) = 120x to find the fourth-order derivative:
f(4)(x) = d^4/dx^4 (f(x)) = d/dx (f(3)(x)) = d/dx (120x) = 120
Therefore, the fourth-order derivative of the function f(x) = 5x^4 is f(4)(x) = 120