MYTUTOR SUBJECT ANSWERS

949 views

Understanding differentiation from first principle.

What is first principle? Why does it look nothing like the differentiation I had been doing for the past few years?

 

Differentiation from first principle is the main idea behind differentiation, a technique we employ to measure instantaneous rate of change. By now, you probably recognize "rate of change" as being synonymous to the term "gradient" or "average speed/distance" in context of several word problems. It is, therefore, not surprising that the definition of first principle is very similar to the gradient formula!

 

Let us recall: the gradient of line connecting two points (x1,y1) and (x2,y2) is given by the equation grad = (y1-y2)/(x1-x2). The definition of first principle says f'(x) = lim_{dx->0} (f(x)-f(x+dx))/dx. (Due to limitation of software, we'll refer to "delta x" by "dx" here for convenience.)

 

Let's examine my claim that both formulas are very similar. Suppose I have a graph y=f(x). Pick a point on the graph and name it (x1,y1). Now, to find the instantaneous rate of change at point (x1,y1), what can we do? Without the knowledge of the equation of the graph, we cannot perform differentiation using the rules that we'd learned. So, instead of trying to get the correct answer, let's approximate. Let's pick a point as cloes to (x1,y1) as we can and name this point (x2,y2). Because these points are very close together, it is safe to say that the line connecting them is a good approximation of the tangent line to point (x1,y1). In that case, we can use the gradient formula to find an approximate solution to the instantaneous gradient at (x1,y1)! By letting the distance between x1 and x2 be dx, we have: x2=x1+dx. Plug this into the gradient formula and we will get

approx. grad = (y1-y2)/(x1-x2) = (f(x1) - f(x2))/(x1 - (x1+dx)) = (f(x1) - f(x1+dx))/dx.

 

There is still something different - the "lim_{dx -> 0}" notation. What does it mean anyway? Reading the symbols out it says "the limit of the expression as "dx" approaches zero". You see, in order to improve accuracy of our gradient's approximation, we only need to pick (x2,y2) to be even closer to (x1,y1). Theoretically, then, if the distance between x1 and x2 is so close that it is almost zero, then our approximation would be the exact solution. Hence, if we allow "dx" to approach zero, then we can confidently change the left-hand-side of the approximation to the exact value, in other words:

f'(x1) = lim_{dx->0} (f(x1)-f(x1+dx))/dx. 

 

Since (x1,y1) is just a name we gave to the point that we are interested in, we may substitute it with (x,y) (to make the formula applicable in a generic x-y plot) to obtain the definition of first principle as presented in our textbooks. 

Jia Hao L. A Level Further Mathematics  tutor, A Level Maths tutor, G...

2 years ago

Answered by Jia Hao, an A Level Further Mathematics tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

101 SUBJECT SPECIALISTS

£20 /hr

Vlad B.

Degree: Mathematics (Masters) - Bath University

Subjects offered:Further Mathematics , Maths

Further Mathematics
Maths

“Hi! I am studying Mathematics at Bath University and I am passionate about teaching, and learning mathematics.”

£24 /hr

Ayusha A.

Degree: BEng electrical and electronics engineering (Bachelors) - Newcastle University

Subjects offered:Further Mathematics , Physics+ 1 more

Further Mathematics
Physics
Maths

“About me: I am a final year Electrical and Electronic Engineering student at Newcastle University. I took Mathematics, Further Mathematics, Chemistry and Physics as my A-level subjects. I did peer mentoring in university and also have...”

£22 /hr

Max G.

Degree: Mathematics and Physics (Masters) - Durham University

Subjects offered:Further Mathematics , Physics+ 1 more

Further Mathematics
Physics
Maths

“I am a maths and physics student at the University of Durham, for as long as I can remember i have been obsessed with all things science! I am patient, friendly and most of all understanding to the fact that the sciences aren't for ev...”

About the author

Jia Hao L.

Currently unavailable: for regular students

Degree: Pure Mathematics and Mathematical Logic (Masters) - Manchester University

Subjects offered:Further Mathematics , Maths

Further Mathematics
Maths

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

You may also like...

Other A Level Further Mathematics questions

Calculate the value of the square root of 3 to four decimal places using the Newton-Raphson process

What is the complex conjugate?

What is the general solution to the equation d2y/dx2 + dy/dx - 2y = -3sinx + cosx (d2y/dx2 signals a second order derivative)

How to determine the rank of a matrix?

View A Level Further Mathematics tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok