High School

Find the indicated derivative, [tex]f^{(4)}(x)[/tex], of the function [tex]f(x) = 5x^6 - 7x^4 + 4x^2[/tex].

Answer :

The indicated derivative of the function f⁴(x) = 1800x² - 168.

The function f(x) = 5x⁶- 7x⁴ + 4x² is a polynomial function. To find its fourth derivative f⁴(x), we need to differentiate it four times using the power rule of differentiation.

1. First Derivative f'(x):

f'(x) = 30x⁵ - 28x³ + 8x

2. Second Derivative f''(x):

f''(x) = 150x⁴ - 84x² + 8

3. Third Derivative f'''(x):

f'''(x) = 600x³ - 168x

4. Fourth Derivative f⁴(x):

f⁴(x) = 1800x² - 168

Thus, the fourth derivative of f(x) = 5x⁶ - 7x⁴ + 4x² is f⁴(x) = 1800x² - 168.