Differentiation concepts J A Rossiter
Introduction The first video introduces the concept of differentiation. What does it mean to differentiate? What is a derivative? What notation is used for differentiation? For now, this series focuses on simple cases, that is functions with a single independent variable and single dependent variable, e.g. y f ( ); z g( w); p v( u)
Gradients It is assumed that by now students understand the term gradient, certainly when applied to a straight 7 line. 6 Gradient is change in dependent variable divided by change in independent variable. 5 4 3 (-,5) - y z y=+ z=-+3 ; 3; (,5) 3 grad y( ) y( ) ; - -3 - -.5 - -.5.5.5.5 3 grad ( (-,-) ) ( ) (3,)
Gradient of a general curve Functions which are not straight lines also have a gradient, but the gradient changes as you move along the curve. y 3 4 Gradient is the instantaneous slope at any given point - - - - -3-3 -4-4- -.5 - -.5.5 - -.5 - -.5.5
What is differentiation? Differentiation is a process which finds the gradient of a curve, precisely, at any point along the curve. 5 y 3 Gradient -.5 at this point. Gradient.75 at this point. - - - - -3-3 Gradient 3.75 at this point. -4-4- -.5 - -.5.5 - -.5 - -.5.5
What is a derivative? A derivative is the result of differentiation, in other words it is the function defining the gradient of a curve, precisely, at any point along the curve. y f ( ) g( Gradient at =-.5 is given by g(-.5) dy d TALKING: Students need to learn how to speak the notation. Gradient at =.5 is - - ) given by g(.5) Notation for a derivative 6 In this case dy/d is spoken as d y d. -3-4 - -.5 - -.5.5
Interim summary Differentiation means to find the gradient; in general this involves some mathematical operations. A derivative is the result of differentiation, that is a function defining the gradient of a curve. The notation of derivative uses the letter d and what looks like a fraction; this is not a fraction and should not be treated as such! y f ( ) w h( z) dy d dw dz derivative derivative dh dz df d Spoken as d f d. Spoken as d h d z. 7
Notation A common mistake for students new to differentiation and derivatives is to assume that one can split up the terms dy and d. YOU CANNOT DO THIS! dy d Is interpreted as single variable or single function. This is specific mathematical notation. Anywhere you see terms like the following, it is most likely they represent derivatives. My suggestion is to avoid using the letter d as a variable and thus avoid any possible confusion with derivative notation. dr ; dt dv ; dh dw ; ds dp ; dr 8
Eamples 9 The following gives eamples of the notation and process, but not detailed derivation which will follow in later videos.
Eample Consider the following function. y 4 3 5 8 6 4 - One can differentiate this function to find the derivative as follows: So, for eample, the gradient of the curve y() at =5 is given as: dy d gradient 4 3 6 at 47-4 3 3.5 4 4.5 5 5.5 6 5 45 3 65
Eample Consider the following function..8.6.4 y t sin( t ). One can differentiate this function to find the derivative as follows: dy dt - -.5.5.5 sin( t) t cos(t) So, for eample, the gradient of the curve y(t) at t= is given as: gradient at sin() cos().77
Other notation At times student will see the notation such as the following: d d d d ( f ( )); ( g( z)); ( a( t)); dz dt This means that the function is to be differentiated, but the result is not yet known. This verbalised as d d of f or d d z of g, etc.
Summary Differentiation means to find the gradient; in general this involves some mathematical operations. A derivative is the result of differentiation, that is a function defining the gradient of a curve. The notation of derivative uses the letter d and is not a fraction! dy df Spoken as y f ( ) derivative d d d f d. d d ( f ) Spoken as d d of f. The action of differentiation. 3
Anthony Rossiter Department of Automatic Control and Systems Engineering University of Sheffield www.shef.ac.uk/acse 6 University of Sheffield This work is licensed under the Creative Commons Attribution. UK: England & Wales Licence. To view a copy of this licence, visit http://creativecommons.org/licenses/by/./uk/ or send a letter to: Creative Commons, 7 Second Street, Suite 3, San Francisco, California 945, USA. It should be noted that some of the materials contained within this resource are subject to third party rights and any copyright notices must remain with these materials in the event of reuse or repurposing. If there are third party images within the resource please do not remove or alter any of the copyright notices or website details shown below the image. (Please list details of the third party rights contained within this work. If you include your institutions logo on the cover please include reference to the fact that it is a trade mark and all copyright in that image is reserved.)