This document summarizes all things to do with nth derivatives/higher order derivatives. It deals with what an nth derivative is and the steps to solve such problems. It includes worked examples with comprehensive step by step notes.
What is the nth Derivative?
If the process of deriving a function is continued 𝑛 times, where 𝑛 is any natural number
(𝑛 ∈ ℕ), then the 𝑛𝑡ℎ derivative can be found which is often referred to as 𝑓 𝑛 (𝑥).
Essentially, the process of finding the 𝑓 𝑛 (𝑥) of a function is determined by deriving the
function a couple of times before establishing a pattern that will work for any 𝑛𝑡ℎ
derivative of the given function.
Steps for Finding the nth Derivative
Step 1: Derive the given function 3 to 4 times, i.e. find the 3rd/4th derivative of the
function. Generally it should only take 3 - 4 derivations to find the pattern.
Step 2: Determine the pattern.
Step 3: Check that the pattern produces the correct results. Substitute in the place of
𝑛, the value 1, 2, 3 or 4 (depends on the last derivative that was derived in step 1) and
check that the answer produced using the pattern matches what was derived for that
function using step 1. This step is simply for checking purposes and therefore does
not form part of the final answer.
Example 1 – Detailed Step by Step
1
Consider the function 𝑓(𝑥) = 𝑥−1
Step 1: Determine 𝑓 ′ (𝑥), 𝑓 2 (𝑥), 𝑓 3 (𝑥) and 𝑓 4 (𝑥)
1 This function can be derived using the quotient or
𝑓(𝑥) =
𝑥−1 chain rule. In this worked example, the chain rule
is applied. In order to use the rule, 𝑥 − 1
∴ 𝑓(𝑥) = (𝑥 − 1)−1 (denominator) is raised to the power of −1.
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller piaeklund. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $2.85. You're not tied to anything after your purchase.
