http://www.ck12.org Chapter 2. Derivatives
The second derivative, f′′, can also be written asddx^2 y 2 , and f′′′can be written asddx^3 y 3. For still higher derivatives,
f(n)=ddxnxn.
Example 10:
Find the fifth derivative off(x) = 2 x^4 − 3 x^3 + 5 x^2 −x−1.
Solution:
f′(x) = 8 x^3 − 9 x^2 + 10 x− 1
f′′(x) = 24 x^2 − 18 x+ 10
f′′′(x) = 48 x− 18
f(^4 )(x) = 48
f(^5 )(x) = 0Example 11:
Show thaty=x^3 + 3 x+2 satisfies the differential equationy′′′+xy′′− 2 y′= 0.
Solution:
We need to obtain the first, second, and third derivatives and substitute them into the differential equation.
y=x^3 + 3 x+ 2
y′= 3 x^2 + 3
y′′= 6 x
y′′′= 6.Substituting,
y′′′+xy′′− 2 y′= 6 +x( 6 x)− 2 ( 3 x^2 + 3 )
= 6 + 6 x^2 − 6 x^2 − 6
= 0which satisfies the equation.
Review Questions
Use the results of this section to find the derivativesdy/dx.
1.y= 5 x^7
2.y=^12 (x^3 − 2 x^2 + 1 )
3.y=√
2 x^3 −√^12 x^2 + 2 x+√
2
4.y=a^2 −b^2 +x^2 −a−b+x(wherea,bare constants)
5.y=x−^3 +x^17