- Topics At a Glance
**Second Derivatives via Formulas**- Third Derivatives and Beyond
- Concavity
- Concave Up
- Concave Down
- No Concavity
- Special Points
- Critical Points
- Points of Inflection
- Extreme Points and How to Find Them
- Finding & Classifying Extreme Points
- First Derivative Test
- Second Derivative Test
- Local vs. Global Points
- Using Derivatives to Draw Graphs
- Finding Points
- Finding Shapes
- Connecting the Dots
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

When we take the derivative of a differentiable function *f*, we end with a new function *f *'.

If this new function *f *' is differentiable, then we can take its derivative to find (f')', also known as *f *" or the *second derivative of f.*

If *f *(*x*) = x^{2} + 4*x*, then we take its derivative once to find

*f *'(*x*) = 2*x* + 4.

Since 2*x* + 4 is a differentiable function, we can take its derivative to find

*f *"(*x*) = 2.

If *f *(*x*) = cos *x* then *f *'(*x*) = -sin *x*. The second derivative of *f* is the derivative of -sin *x*, so*f *"(*x*) = -cos *x*.

Not all continuous functions are differentiable, and not all differentiable functions have second derivatives.

Example. Look at the piecewise-defined function

This function is continuous, since it is continuous on each of the pieces and since

It is also differentiable. We can take the derivative of each piece to find

This new function *f *' is the absolute value function. We could write

*f *'(*x*) = | *x *|.

We know that the absolute value function is not differentiable, which means we can't take another derivative here. The second derivative of *f*, also known as *f *"(*x*), does not exist.

Exercise 1

Find the second derivative of the function.

*f *(*x*) = sin *x*

Exercise 2

Find the second derivative of the function.

*f*(*x*) = 8*x*^{4} + 3*x*^{2} - 9*x*

Exercise 3

Find the second derivative of the function.

*g*(*x*) = *x* + 7

Exercise 4

Find the second derivative of the function.

*g*(*x*) = *ln* *x*

Exercise 5

Find the second derivative of the function.

*h*(*x*) = 2^{x}

Exercise 6

For the function, either find the second derivative or determine that the second derivative does not exist.

*f*(*x*) = *xe*^{x}

Exercise 7

For the function, either find the second derivative or determine that the second derivative does not exist.

*f*(*x*) = 8

Exercise 8

For the function, either find the second derivative or determine that the second derivative does not exist.

*f*(*x*) = *x*^{(5/3)}

Exercise 9

*f*(*x*) = |*x*|

Exercise 10

*f*(*x*) = *e*^{x}