Determine the second derivative for following functions.

                            Q (t ) = sec (5t )

Solution : Following is the first derivative.

             Q′ (t ) = 5 sec (5t ) tan (5t )

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

Q′′ (t ) = 25 sec (5t ) tan (5t ) tan (5t ) + 25 sec (5t ) sec² (5t )

            = 25 sec (5t ) tan 2 (5t ) + 25 sec³ (5t )

Notice  as well that each successive derivative will needs a product and/or chain rule & that as noted above it will not ending up returning back to only a secant after four (or another other number for that matter) derivatives as sine & cosine will.

 As we saw previously we will frequently need to utilize the product or quotient rule for the higher order derivatives, even while the first derivative didn't need these rules.

Let's work one more instance that will see how to utilize implicit differentiation to determine higher order derivatives.