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Finding the Gradient of a Curve and Understanding Stationary Points

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Millie Roberts

14/09/2025

Maths

Differentiation Q and A

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Maths

Algebra

Straight line graphs

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78

•

14 Sept 2025

•

3 pages

Finding the Gradient of a Curve and Understanding Stationary Points

user profile picture

Millie Roberts

@millierobertsz

Learn how to find the gradient of a curve using dy/dx and calculate it at points like x=2. Discover how to find the gradient of a function f(x, y, z) and explore tangents. Understand stationary points and how to determine if they're maximum or minimum using the second derivative. Get examples and use calculators for finding stationary points and gradients. Perfect for kids curious about calculus and PDF resources!

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1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Finding Stationary Points

This page focuses on finding the coordinates of stationary points and determining their nature. It continues the example from the previous page and introduces a new, more complex problem.

For the function y = 3x + 1/x², the page demonstrates how to find stationary points by setting dy/dx = 0 and solving for x.

Example: For y = 3x + 1/x², the stationary points occur at x = ±1/√3.

The page then introduces a new problem: y = 4x³ + 15x² - 18x + 5. It guides the reader through the process of finding the stationary points for this function.

Highlight: Stationary points occur where dy/dx = 0, which is crucial for finding the gradient of a tangent at these points.

Vocabulary: Stationary points are points on a curve where the gradient is zero or undefined.

This section provides valuable insight into how to determine nature of stationary points using second derivative, which is explored further on the next page.

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Analyzing Stationary Points

The final page delves deeper into the analysis of stationary points, focusing on determining their nature using the second derivative test.

For the function y = 4x³ + 15x² - 18x + 5, the page shows how to calculate the second derivative (d²y/dx²) and use it to classify the stationary points.

Example: For the stationary point at x = 1/2, d²y/dx² > 0, indicating a minimum point. For x = -3, d²y/dx² < 0, indicating a maximum point.

Definition: The second derivative test states that if d²y/dx² > 0 at a stationary point, it's a minimum; if d²y/dx² < 0, it's a maximum.

This page effectively demonstrates how to determine nature of stationary points using second derivative, providing a comprehensive understanding of curve analysis.

Highlight: Understanding the nature of stationary points is crucial for graphing functions and solving optimization problems in calculus.

The document concludes by emphasizing the importance of these techniques in analyzing functions and their behavior, which is essential for advanced mathematical studies and applications.

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Finding the Gradient of a Curve

This page introduces methods for finding the gradient of a curve using dy/dx. It presents two main examples to illustrate the process.

The first example deals with the function y = 2x³ + 5x² - 7x + 10. The page demonstrates how to find dy/dx and then calculate the gradient at a specific point.

Example: For y = 2x³ + 5x² - 7x + 10, dy/dx = 6x² + 10x - 7. When x = 2, the gradient is 37.

The second example involves a more complex function, y = 3x + 1/x². The page shows how to differentiate this function and find its gradient.

Highlight: The differentiation rule for x^n is nx^n−1n-1, which is crucial for finding dy/dx.

Vocabulary: dy/dx represents the derivative of y with respect to x, which gives the gradient function of the curve.

This page provides a solid foundation for understanding how to find the gradient of a curve at a point where x=2 or any other given x-value.



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Where can I download the Knowunity app?

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4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

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

iOS user

Best app on earth! no words because it’s too good

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

iOS user

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

iOS user

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

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

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

iOS user

Best app on earth! no words because it’s too good

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

iOS user

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

iOS user

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

iOS user

Subjects

Maths

Algebra

Straight line graphs

Coordinates

 

Maths

•

78

•

14 Sept 2025

•

3 pages

Finding the Gradient of a Curve and Understanding Stationary Points

user profile picture

Millie Roberts

@millierobertsz

Learn how to find the gradient of a curve using dy/dx and calculate it at points like x=2. Discover how to find the gradient of a function f(x, y, z) and explore tangents. Understand stationary points and how to determine if they're maximum or minimum using the second derivative. Get examples and use calculators for finding stationary points and gradients. Perfect for kids curious about calculus and PDF resources!

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Finding Stationary Points

This page focuses on finding the coordinates of stationary points and determining their nature. It continues the example from the previous page and introduces a new, more complex problem.

For the function y = 3x + 1/x², the page demonstrates how to find stationary points by setting dy/dx = 0 and solving for x.

Example: For y = 3x + 1/x², the stationary points occur at x = ±1/√3.

The page then introduces a new problem: y = 4x³ + 15x² - 18x + 5. It guides the reader through the process of finding the stationary points for this function.

Highlight: Stationary points occur where dy/dx = 0, which is crucial for finding the gradient of a tangent at these points.

Vocabulary: Stationary points are points on a curve where the gradient is zero or undefined.

This section provides valuable insight into how to determine nature of stationary points using second derivative, which is explored further on the next page.

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Analyzing Stationary Points

The final page delves deeper into the analysis of stationary points, focusing on determining their nature using the second derivative test.

For the function y = 4x³ + 15x² - 18x + 5, the page shows how to calculate the second derivative (d²y/dx²) and use it to classify the stationary points.

Example: For the stationary point at x = 1/2, d²y/dx² > 0, indicating a minimum point. For x = -3, d²y/dx² < 0, indicating a maximum point.

Definition: The second derivative test states that if d²y/dx² > 0 at a stationary point, it's a minimum; if d²y/dx² < 0, it's a maximum.

This page effectively demonstrates how to determine nature of stationary points using second derivative, providing a comprehensive understanding of curve analysis.

Highlight: Understanding the nature of stationary points is crucial for graphing functions and solving optimization problems in calculus.

The document concludes by emphasizing the importance of these techniques in analyzing functions and their behavior, which is essential for advanced mathematical studies and applications.

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Finding the Gradient of a Curve

This page introduces methods for finding the gradient of a curve using dy/dx. It presents two main examples to illustrate the process.

The first example deals with the function y = 2x³ + 5x² - 7x + 10. The page demonstrates how to find dy/dx and then calculate the gradient at a specific point.

Example: For y = 2x³ + 5x² - 7x + 10, dy/dx = 6x² + 10x - 7. When x = 2, the gradient is 37.

The second example involves a more complex function, y = 3x + 1/x². The page shows how to differentiate this function and find its gradient.

Highlight: The differentiation rule for x^n is nx^n−1n-1, which is crucial for finding dy/dx.

Vocabulary: dy/dx represents the derivative of y with respect to x, which gives the gradient function of the curve.

This page provides a solid foundation for understanding how to find the gradient of a curve at a point where x=2 or any other given x-value.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.

Where can I download the Knowunity app?

You can download the app from Google Play Store and Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Smart Tools NEW

Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines

Mock Exam
Quiz
Flashcards
Essay

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

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

iOS user

Best app on earth! no words because it’s too good

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

iOS user

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

iOS user

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

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

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

iOS user

Best app on earth! no words because it’s too good

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

iOS user

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

iOS user

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

iOS user