Interview
The first principles
- Course
- Institution
Differentiation from the first principles
[Show more]
Seller
Follow
kalebroodt
Content preview
Definition of a derivative:What is a derivative?
The derivative of a function measures the rate at which the dependent variable changes as the
independent variable changes.
Qué?
In the Cartesian plane we use 𝑦 to represent the dependent variable, and 𝑥 to represent the
independent variable.
So, the derivative of a function measures the rate at which 𝒚 changes as 𝒙 changes.
In case you feel like you are suffering from a touch of déjà vu… “the rate at which 𝑦 changes as
𝑥 changes” we are indeed familiar with; we call it the gradient of a function.
In a Nutshell: The value of a derivative at a particular point is the same as the gradient of the
tangent to the curve at that point.
∴ 𝒎 = 𝒇′(𝒙)
𝑓′(𝑥) is one of the notations we use for the derivative… this reads as “the derivative of 𝑓 with
respect to 𝑥” or “𝑓 prime of 𝑥”, and is often referred to informally as “f dash”.
And we just so happen to have an equation for the gradient of the tangent to a curve:
𝑚𝑎𝑣𝑒
𝑓(𝑥+∆𝑥)−𝑓(𝑥) where ∆𝑥 → 0
= ∆𝑥
First thing we are going to do is make a simple substitution. Instead of ∆𝒙 being the change in the
𝑥- value, now we are going to represent the 𝑥-increment between the two points with a simple
variable instead… we will call it 𝒉… this is done for two reasons:
1. It is international standard practice
2. It will vastly simplify the appearance of the algebra. (you’ll thank me for this…)
So now our equation for the average gradient at a point on a curve is:
𝑚𝑎𝑣𝑒 𝑓(𝑥+ℎ)−𝑓(𝑥)
= ℎ where ℎ → 0
ℎ
ℎ
ℎ
ℎ
ℎ
ℎ
As the value of ℎ → 0, the gradient of the resultant secant better approximates the gradient of the
1
, tangent to the curve at point A.
2
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
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 kalebroodt. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $3.98. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
70055 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now