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12/16/2025
Maths
Differentiation - Higher Maths
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Dec 16, 2025
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Mastering Differentiation: Higher Maths Practice
S
sy7
@sy7_quyl
1 / 5
Cubic Functions and Stationary Points
You'll often encounter cubic functions like f(x) = x³ - 6x² + 9x in your exams. The key is finding where the curve has maximum and minimum points by setting f'(x) = 0.
When you differentiate this function, you get f'(x) = 3x² - 12x + 9. Setting this equal to zero and solving gives you the x-coordinates of your stationary points. Remember that maximum points have f''(x) < 0, whilst minimum points have f''(x) > 0.
Quick Tip: Always check your stationary points using the second derivative test - it's the fastest way to determine their nature!
The graph shows a classic cubic shape with one maximum at point A and one minimum at point B(3, 0). This pattern appears frequently in exam questions, so get comfortable with identifying these features.
Derivative Graphs and Gradient Calculations
Understanding how to sketch derivative graphs from the original function is crucial for Paper 1. When the original function has a maximum, f'(x) crosses from positive to negative through zero. At minimum points, f'(x) crosses from negative to positive.
For gradient calculations, remember that finding f'(4) when f(x) = √x + 2/x² means differentiating first, then substituting. You'll get f'(x) = 1/(2√x) - 4/x³, so f'(4) = 1/4 - 4/64 = 1/4 - 1/16.
Tangent lines are another favourite exam topic. When the gradient equals a specific value , you set the derivative equal to that value: 12x - 3x² = 12. Solve for x, then find the corresponding y-coordinate.
Remember: The derivative at any point gives you the gradient of the tangent line at that point.
Advanced Applications and Turning Points
Turning points questions often involve finding coordinates and determining their nature. For y = x³ - 3x² - 9x + 12, you'd differentiate to get y' = 3x² - 6x - 9, then solve 3x² - 6x - 9 = 0.
Composite functions like p(x) = f(g(x)) require the chain rule for differentiation. If f(x) = 3x + 1 and g(x) = x² - 2, then p(x) = 3 + 1 and p'(x) = 6x.
Circle and parabola problems combine geometry with calculus. When a circle touches a parabola at two points, the tangent gradients at those points are parallel to the line joining the circle's centre to each point.
Pro Tip: Always expand and simplify your functions before differentiating - it makes the algebra much easier!
Maximum and minimum value problems on closed intervals require checking both stationary points and endpoints. For f(x) = x³ - 2x² - 4x + 6 on , evaluate f at x = 0, x = 3, and any stationary points in between.
Complex Differentiation and Applications
Higher-order polynomials like y = x⁴ + 4x³ + 2x² - 20x + 3 can be tricky, but the method stays the same. Differentiate to find y' = 4x³ + 12x² + 4x - 20, then solve y' = 0. Sometimes you'll find only one stationary point, which you can verify using the discriminant.
Trigonometric differentiation appears in mechanics problems. When velocity v(t) = 8cos, acceleration a(t) = v'(t) = -16sin. The chain rule is essential here because of the inside the cosine.
Fractional and root functions need careful handling. For f(x) = x√x - 3x - 2/(x√x), rewrite using indices first: f(x) = x^(3/2) - 3x - 2x^(-3/2). Then differentiate term by term.
Key Strategy: Always rewrite roots and fractions as powers before differentiating - it prevents silly mistakes.
For increasing functions, you need f'(x) > 0. After finding stationary points, test the sign of f'(x) in each interval to determine where the function is strictly increasing.
Final Tips and Common Mistakes
Practice makes perfect with differentiation past papers. Focus on recognising question types quickly, and always double-check your algebra when finding stationary points.
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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.
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Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Mastering Differentiation: Higher Maths Practice
S
sy7
@sy7_quyl
Differentiation is one of the most important topics in A-level maths, and these past paper questions show you exactly what examiners love to test. From finding gradients and stationary points to sketching curves and working with tangent lines, these problems... Show more
Cubic Functions and Stationary Points
You'll often encounter cubic functions like f(x) = x³ - 6x² + 9x in your exams. The key is finding where the curve has maximum and minimum points by setting f'(x) = 0.
When you differentiate this function, you get f'(x) = 3x² - 12x + 9. Setting this equal to zero and solving gives you the x-coordinates of your stationary points. Remember that maximum points have f''(x) < 0, whilst minimum points have f''(x) > 0.
Quick Tip: Always check your stationary points using the second derivative test - it's the fastest way to determine their nature!
The graph shows a classic cubic shape with one maximum at point A and one minimum at point B(3, 0). This pattern appears frequently in exam questions, so get comfortable with identifying these features.
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Derivative Graphs and Gradient Calculations
Understanding how to sketch derivative graphs from the original function is crucial for Paper 1. When the original function has a maximum, f'(x) crosses from positive to negative through zero. At minimum points, f'(x) crosses from negative to positive.
For gradient calculations, remember that finding f'(4) when f(x) = √x + 2/x² means differentiating first, then substituting. You'll get f'(x) = 1/(2√x) - 4/x³, so f'(4) = 1/4 - 4/64 = 1/4 - 1/16.
Tangent lines are another favourite exam topic. When the gradient equals a specific value , you set the derivative equal to that value: 12x - 3x² = 12. Solve for x, then find the corresponding y-coordinate.
Remember: The derivative at any point gives you the gradient of the tangent line at that point.
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Advanced Applications and Turning Points
Turning points questions often involve finding coordinates and determining their nature. For y = x³ - 3x² - 9x + 12, you'd differentiate to get y' = 3x² - 6x - 9, then solve 3x² - 6x - 9 = 0.
Composite functions like p(x) = f(g(x)) require the chain rule for differentiation. If f(x) = 3x + 1 and g(x) = x² - 2, then p(x) = 3 + 1 and p'(x) = 6x.
Circle and parabola problems combine geometry with calculus. When a circle touches a parabola at two points, the tangent gradients at those points are parallel to the line joining the circle's centre to each point.
Pro Tip: Always expand and simplify your functions before differentiating - it makes the algebra much easier!
Maximum and minimum value problems on closed intervals require checking both stationary points and endpoints. For f(x) = x³ - 2x² - 4x + 6 on , evaluate f at x = 0, x = 3, and any stationary points in between.
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Complex Differentiation and Applications
Higher-order polynomials like y = x⁴ + 4x³ + 2x² - 20x + 3 can be tricky, but the method stays the same. Differentiate to find y' = 4x³ + 12x² + 4x - 20, then solve y' = 0. Sometimes you'll find only one stationary point, which you can verify using the discriminant.
Trigonometric differentiation appears in mechanics problems. When velocity v(t) = 8cos, acceleration a(t) = v'(t) = -16sin. The chain rule is essential here because of the inside the cosine.
Fractional and root functions need careful handling. For f(x) = x√x - 3x - 2/(x√x), rewrite using indices first: f(x) = x^(3/2) - 3x - 2x^(-3/2). Then differentiate term by term.
Key Strategy: Always rewrite roots and fractions as powers before differentiating - it prevents silly mistakes.
For increasing functions, you need f'(x) > 0. After finding stationary points, test the sign of f'(x) in each interval to determine where the function is strictly increasing.
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
Final Tips and Common Mistakes
Practice makes perfect with differentiation past papers. Focus on recognising question types quickly, and always double-check your algebra when finding stationary points.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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.
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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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user