Higher Derivatives

1. Higher Derivatives The first time that a function is differentiated, the result is called the first derivative. If this derivative is differentiated again, the result is called the second derivative. This process can be repeated as long as it continues to be possible to differentiate. These derivatives (beyond the first derivative) are known as higher derivatives

2. Notations for Higher Derivatives • Second Derivative: f" (x) or y " or • Third derivative: f"' (x) or y "' or • These patterns continue as long as it is possible.

3. Meanings of Derivatives • The first derivative represents the instantaneous rate of change of a function. • That is the slope of the tangent to the function at a point. • The second derivative represents the rate at which the slope of the tangents to the function are changing as we move through the domain of the function

4. Polynomial Functions EXAMPLE 1: Ify = x3 – 2x2 + 5x – 7then Notice the change in degree from cubic to quadratic to linear. This pattern of reduction in degree is always true with polynomial functions, but the pattern becomes more complex with rational functions.

5. EXAMPLE 2: Find the second derivative of First Derivative Second Derivative EXAMPLE 3:Find the third derivative off(x) = – 2x3 – 4x2 + 11x – 3 Second Derivative First Derivative f"(x) = –12x– 8 f′ (x) = – 6x2 – 8x + 11 Third Derivative f"'(x) = –12

6. EXAMPLE 4: For the functionf(x) = 12x4 – 3x2, find f " (–2). f ′ (x) = 48x3 – 3x f"(x) = 144x2 – 6 f " (–2) = 144(–2)2– 6 = 570

7. EXAMPLE 5:For f(x) = 4x3 – x + 3 find f " (–1)

8. Chain Rule on Polynomial Function EXAMPLE 6: Find the second derivative ofy = (x 3 + 2)5 Second Derivative With Product Rule First Derivative

9. EXAMPLE 7: Find the second derivative ofy = (2x 2 – 13)4 Second Derivative Use Product Rule First Derivative

10. EXAMPLE 8: Find the second derivative of Second Derivative First Derivative OR OR

11. Implicit Differentiation and Higher Order Derivatives EXAMPLE 9: Find if