What is exactly differentiation?

This is obviously a very important but somewhat difficult to explain question of maths. Let's try to define these terms for normal functions between R (real numbers) and R. 
Now, if we consider the derivative of f at a certain point (let's say x), you can think of it as looking at the gradient of f at that point. So, if you're function f is constant, then we have a flat line, and so we have that it's gradient everywhere is 0, and therefore it's derivative is zero. (i.e. f'(x) = 0 for all x). If, however we have a linear function such as f(x) = 2x + 1, if we look at the graph we see that it's gradient is 2 (using the simple gradient formula), hence f'(x) = 2. 
Of course we don't have to use the gradient formula every time, sometimes we won't even be able to (when the function isn't linear, that is), and there is a very helpful rule for functions of the type f(x) = x^n. 
That is, f'(x) = nx^(n-1). 
But it is still good to understand what the derivative really is, once you understand it's relationship with the gradient, you will already be ahead of most other A-level mathematicians. Indeed, have you ever wondered why f has a minimum at x if f'(x) = 0? Draw a picture of f and what it looks like when the gradient is 0 at a certain point, and it will all become natural. 

JP
Answered by Joseph P. Maths tutor

5210 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Solve x^3+2x^2+x=0


I can differentiate exponentials (e^x), but how can I differentiate ln(x)?


Find dy/dx in terms of t for the curve given by the parametric equations x = tan(t) , y = sec(t) for -pi/2<t<pi/2.


How would I answer this question? Use factor theorem to show (x-2) is a factor of f(x) = 2x^3 -7x^2 +4x +4.


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences