### Home > CALC > Chapter 5 > Lesson 5.1.1 > Problem 5-7

You know that the first derivative,

*f*′(*x*), tells us the slope and the rate of change of*f*(*x*). Homework Help ✎What does the

*second derivative*,*f*″(*x*), tell you about*f*′(*x*)? What does*f*″(*x*) tell you about*f*(*x*)?Find

*f*′(*x*) and*f*″(*x*) for*f*(*x*) =*x*^{3}+ 3*x*^{2}− 9*x*+ 2.Use these derivatives to find

*f*″( l ) and*f*″(−2). Is*f*(*x*) getting steeper or less steep at*x*= 1? At*x*= −2? Explain your reasoning.The values in part (c) can be used to determine concavity. Where is

*f*(*x*) concave up? Where is*f*(*x*) concave down?

*f* ''(*x*) tells us the same thing about *f* '(*x*) that *f* '(*x*) tells us about *f*(*x*).

*f* '(*x*): 3*x*² + 6*x* −9*f* ''(*x*): 6*x* + 6

Consider this: Even though a slope of −2 is steeper than a slope of −1, if the slope changes from −1 to −2, then the slope is decreasing.

Positive values of the 2nd-derivative indicate that the function is concave up while negative values indicate that the function is concave down.