Higher Order Derivatives and Practice Problems | Calculus - Mathematics PDF Download

Notice that differentiating an exponential function is very simple. It doesn’t change with each differentiation.
(b) y = cos x
Again, let’s just do some derivatives.

Note that cosine (and sine) will repeat every four derivatives. The other four trig functions will not exhibit this behavior. You might want to take a few derivatives to convince yourself of this.
(c)
In the previous two examples we saw some patterns in the differentiation of exponential functions, cosines and sines. We need to be careful however since they only work if there is just a t t or an x x in the argument. This is the point of this example. In this example we will need to use the chain rule on each derivative.

So, we can see with slightly more complicated arguments the patterns that we saw for exponential functions, sines and cosines no longer completely hold.
Let’s do a couple more examples to make a couple of points.
Example 2 Find the second derivative for each of the following functions.
(a)
(b)
(c)
Solution: (a)
Here’s the first derivative. 

Notice that the second derivative will now require the product rule.

Notice that each successive derivative will require a product and/or chain rule and that as noted above this will not end up returning back to just a secant after four (or another other number for that matter) derivatives as sine and cosine will.
(b)
Again, let’s start with the first derivative.

As with the first example we will need the product rule for the second derivative.

(c)

Same thing here.

The second derivative this time will require the quotient rule.

As we saw in this last set of examples we will often need to use the product or quotient rule for the higher order derivatives, even when the first derivative didn’t require these rules.

Let’s work one more example that will illustrate how to use implicit differentiation to find higher order derivatives.

Example 3 Find y′′for x² + y⁴ = 10 
Solution: Okay, we know that in order to get the second derivative we need the first derivative and in order to get that we’ll need to do implicit differentiation. Here is the work for that.

Now, this is the first derivative. We get the second derivative by differentiating this, which will require implicit differentiation again.

This is fine as far as it goes. However, we would like there to be no derivatives in the answer. We don’t, generally, mind having x x’s and/or y y’s in the answer when doing implicit differentiation, but we really don’t like derivatives in the answer. We can get rid of the derivative however by acknowledging that we know what the first derivative is and substituting this into the second derivative equation. Doing this gives,

 Now that we’ve found some higher order derivatives we should probably talk about an interpretation of the second derivative. If the position of an object is given by s ( t ) we know that the velocity is the first derivative of the position.

The acceleration of the object is the first derivative of the velocity, but since this is the first derivative of the position function we can also think of the acceleration as the second derivative of the position function.

Alternate Notation
There is some alternate notation for higher order derivatives as well. Recall that there was a fractional notation for the first derivative.

We can extend this to higher order derivatives.
 etc.
Practice Problems
Question 1: Determine the fourth derivative of h ( t ) = 3 t 7 − 6 t 4 + 8 t 3 − 12 t + 18
Solution: Step 1 Not much to this problem other than to take four derivatives so each step will show each successive derivative until we get to the fourth. The first derivative is then,

Step 2 The second derivative is, 
Step 3 The third derivative is, 
Step 4 The fourth, and final derivative for this problem, is,
Question 2 Determine the fourth derivative of
Solution: Step 1 Not much to this problem other than to take four derivatives so each step will show each successive derivative until we get to the fourth. The first derivative is then,

Step 2 The second derivative is,
Step 3 The third derivative is,

Step 4 The fourth, and final derivative for this problem, is,
V⁽⁴⁾ (x) = 0
Note that we could have just as easily used the Fact from the notes to arrive at this answer in one step. 
Question 3 Determine the fourth derivative of  
Solution: Step 1 Not much to this problem other than to take four derivatives so each step will show each successive derivative until we get to the fourth. After a quick rewrite of the function to help with the differentiation the first derivative is,
  
Step 2 The second derivative is, 
Step 3 The third derivative is, 
Step 4 The fourth, and final derivative for this problem, is,
Question 4 Determine the fourth derivative of f ( w ) = 7 sin ( w/3) + cos ( 1 − 2 w )
Solution: Step 1 Not much to this problem other than to take four derivatives so each step will show each successive derivative until we get to the fourth. The first derivative is then, 

Step 2 The second derivative is,
Step 3 The third derivative is, 
Step 4 The fourth, and final derivative for this problem, is,

Question 5 Determine the fourth derivative of
Solution: Step 1 Not much to this problem other than to take four derivatives so each step will show each successive derivative until we get to the fourth. The first derivative is then, 

Step 2 The second derivative is, 
Step 3 The third derivative is,

Step 4 The fourth, and final derivative for this problem, is, Question Question 6 Determine the second derivative of
Solution: Not much to this problem other than to take two derivatives so each step will show each successive derivative until we get to the second. The first derivative is then,

Do not forget that often we will end up needing to do a product rule in the second derivative even though we did not need to do that in the first derivative. The second derivative is then,

Question 7 Determine the second derivative of z=ln(7−x³) 
Solution: Step 1 Not much to this problem other than to take two derivatives so each step will show each successive derivative until we get to the second. The first derivative is then,

Step 2 Do not forget that often we will end up needing to do a quotient rule in the second derivative even though we did not need to do that in the first derivative. The second derivative is then,

Question 8 Determine the second derivative of Q(v)= 2/(6+2v - v²)⁴
Solution: Step 1 Not much to this problem other than to take two derivatives so each step will show each successive derivative until we get to the second. We’ll do a quick rewrite of the function to help with the derivatives and then the first derivative is,

Do not forget that often we will end up needing to do a product rule in the second derivative even though we did not need to do that in the first derivative. The second derivative is then, 

Question 9 Determine the second derivative of H ( t ) = cos 2 ( 7 t )
Solution: Step 1 Not much to this problem other than to take two derivatives so each step will show each successive derivative until we get to the second. The first derivative is then,
 
Step 2 Do not forget that often we will end up needing to do a product rule in the second derivative even though we did not need to do that in the first derivative. The second derivative is then,

Note that, in this case, if we recall our trig formulas we could have reduced the product in the first derivative to a single trig function which would have then allowed us to avoid the product rule for the second derivative. Can you figure out what the formula is? 
Question 10 Determine the second derivative of 
Solution: Step 1 Not much to this problem other than to take two derivatives so each step will show each successive derivative until we get to the second. Note however that we are going to have to do implicit differentiation to do each derivative.

Here is the work for the first derivative. If you need a refresher on implicit differentiation go back to that section and check some of the problems in that section.

   
Step 2 Now, the second derivative will also need implicit differentiation. Note as well that we can work with the first derivative in its present form which will require the quotient rule or we can rewrite it as,  
Step 3 Finally, recall that we don’t want a y ′ y′ in the second derivative so to finish this out we need to plug in the formula for y ′ y′ (which we know…) and do a little simplifying to get the final answer.