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Understanding Differentiation: Gradients, Tangents, and More!

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Kate

15/09/2023

Maths

AQA Pure Math- Differentiation- Year 1 and 2

Understanding Differentiation: Gradients, Tangents, and More!

A comprehensive guide to first principles of differentiation explained, covering fundamental calculus concepts and advanced differentiation techniques.

  • The guide explores differentiation from first principles, including the limit definition and practical applications
  • Covers essential topics like gradient and tangents in calculus, increasing/decreasing functions, and stationary points
  • Details various differentiation rules including chain rule, product rule, and implicit differentiation in calculus
  • Examines trigonometric differentiation and special functions like exponentials and logarithms
  • Includes practical examples and step-by-step solutions for complex differentiation problems
...

15/09/2023

210

C
Differentiation
.
Differentiation
y = f (oc)
dy = f (x)
doc
• First Principles
f'(x) = lim f(x+h)-f(x)
[→]
h
as h tends to zero'
example
p

View

Page 2: Tangents, Normals, and Function Behavior

This page delves into the properties of tangents and normals, along with the analysis of increasing and decreasing functions. It also covers second derivatives and their applications in finding turning points.

Definition: A tangent has the same gradient as the curve at the point of contact, while a normal is perpendicular to the tangent

Highlight: Second derivatives f"(xf"(x) are crucial for determining the nature of turning points and curve behavior

Example: For a quadratic function fxx = ax² + bx + c, where a > 0, the second derivative helps identify maximum and minimum points

Vocabulary: Stationary point - a point where the gradient equals zero

C
Differentiation
.
Differentiation
y = f (oc)
dy = f (x)
doc
• First Principles
f'(x) = lim f(x+h)-f(x)
[→]
h
as h tends to zero'
example
p

View

Page 3: Advanced Differentiation Rules

This page covers more complex differentiation techniques, including trigonometric functions, exponentials, and the chain rule. It provides comprehensive examples of various differentiation methods.

Definition: Chain rule states that if y is a function of u, which is a function of x, then dy/dx = dy/du × du/dx

Example: Differentiating y = ekx results in dy/dx = kekx, demonstrating the application of the chain rule

Highlight: The product rule is essential for differentiating products of functions: d/dxuvuv = udv/dxdv/dx + vdu/dxdu/dx

Vocabulary: Product rule - a method for differentiating the product of two functions

C
Differentiation
.
Differentiation
y = f (oc)
dy = f (x)
doc
• First Principles
f'(x) = lim f(x+h)-f(x)
[→]
h
as h tends to zero'
example
p

View

Page 4: Quotient Rule and Implicit Differentiation

This page explains the quotient rule and implicit differentiation, including applications to trigonometric functions and complex expressions.

Definition: Quotient rule states that d/dxu/vu/v = v(du/dxv(du/dx - udv/dxdv/dx)/v²

Example: Differentiating y = x2+1x²+1/x3x-3 using the quotient rule demonstrates practical application

Highlight: Implicit differentiation is crucial when dealing with equations where y cannot be easily isolated

Vocabulary: Implicit differentiation - a method used when y cannot be explicitly expressed in terms of x

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Maths

210

10 Jul 2025

4 pages

Understanding Differentiation: Gradients, Tangents, and More!

user profile picture

Kate

@katerevisionotes

A comprehensive guide to first principles of differentiation explained, covering fundamental calculus concepts and advanced differentiation techniques.

  • The guide explores differentiation from first principles, including the limit definition and practical applications
  • Covers essential topics like gradient and tangents in... Show more

C
Differentiation
.
Differentiation
y = f (oc)
dy = f (x)
doc
• First Principles
f'(x) = lim f(x+h)-f(x)
[→]
h
as h tends to zero'
example
p

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Page 2: Tangents, Normals, and Function Behavior

This page delves into the properties of tangents and normals, along with the analysis of increasing and decreasing functions. It also covers second derivatives and their applications in finding turning points.

Definition: A tangent has the same gradient as the curve at the point of contact, while a normal is perpendicular to the tangent

Highlight: Second derivatives f"(xf"(x) are crucial for determining the nature of turning points and curve behavior

Example: For a quadratic function fxx = ax² + bx + c, where a > 0, the second derivative helps identify maximum and minimum points

Vocabulary: Stationary point - a point where the gradient equals zero

C
Differentiation
.
Differentiation
y = f (oc)
dy = f (x)
doc
• First Principles
f'(x) = lim f(x+h)-f(x)
[→]
h
as h tends to zero'
example
p

Sign up to see the contentIt's free!

Access to all documents

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By signing up you accept Terms of Service and Privacy Policy

Page 3: Advanced Differentiation Rules

This page covers more complex differentiation techniques, including trigonometric functions, exponentials, and the chain rule. It provides comprehensive examples of various differentiation methods.

Definition: Chain rule states that if y is a function of u, which is a function of x, then dy/dx = dy/du × du/dx

Example: Differentiating y = ekx results in dy/dx = kekx, demonstrating the application of the chain rule

Highlight: The product rule is essential for differentiating products of functions: d/dxuvuv = udv/dxdv/dx + vdu/dxdu/dx

Vocabulary: Product rule - a method for differentiating the product of two functions

C
Differentiation
.
Differentiation
y = f (oc)
dy = f (x)
doc
• First Principles
f'(x) = lim f(x+h)-f(x)
[→]
h
as h tends to zero'
example
p

Page 4: Quotient Rule and Implicit Differentiation

This page explains the quotient rule and implicit differentiation, including applications to trigonometric functions and complex expressions.

Definition: Quotient rule states that d/dxu/vu/v = v(du/dxv(du/dx - udv/dxdv/dx)/v²

Example: Differentiating y = x2+1x²+1/x3x-3 using the quotient rule demonstrates practical application

Highlight: Implicit differentiation is crucial when dealing with equations where y cannot be easily isolated

Vocabulary: Implicit differentiation - a method used when y cannot be explicitly expressed in terms of x

C
Differentiation
.
Differentiation
y = f (oc)
dy = f (x)
doc
• First Principles
f'(x) = lim f(x+h)-f(x)
[→]
h
as h tends to zero'
example
p

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

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Page 1: First Principles and Basic Differentiation

This page introduces the fundamental concepts of differentiation, starting with the first principle formula and its applications. The content focuses on basic differentiation rules and gradient calculations.

Definition: First principle of differentiation is defined as f'xx = limh0h→0 f(x+h)f(x)f(x+h)-f(x)/h

Example: Proving the first principle of 5x³ results in 15x², demonstrating the practical application of the limit definition

Highlight: Understanding gradient at a point is crucial for analyzing curve behavior, as shown in the example where gradient at point 2,22,2 is calculated

Vocabulary: Derivative f(xf'(x) - the rate of change of a function at any given point

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This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

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Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

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The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

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This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

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