Question

For the function f(x)=3 x⁵-4 x⁴+3 x³+5 x²+4, find fʹ''(x), the third derivative of f, and f⁴(x), the fourth derivative of f.

Answer 1

The third derivative of the function f(x) is f′′′(x) = 180x² - 96x + 18, and the fourth derivative is f⁴(x) = 360x - 96.

Step-by-step explanation:

To find the third derivative of the function f(x) = 3x⁵ - 4x⁴ + 3x³ + 5x² + 4, we successively differentiate the function with respect to x. The first derivative, or f′(x), the second derivative, or f′′(x), and finally the third derivative, or f′′′(x), are computed as follows:

• f′(x) = 15x⁴ - 16x³ + 9x² + 10x
• f′′(x) = 60x³ - 48x² + 18x + 10
• f′′′(x) = 180x² - 96x + 18

The fourth derivative of the function, or f⁴(x), is obtained by differentiating the third derivative:

• f⁴(x) = 360x - 96

After the fourth derivative, all further derivatives will be constants since the terms remaining are only x to the first power or constants.