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Mastering Differentiation: Higher Maths Practice
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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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Samantha Klich
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Thomas R
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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
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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
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
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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 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.
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Most popular content: Differentiation
Most popular content in Maths
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Can't find what you're looking for? Explore other subjects.
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4.9/5
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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