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Saturday, 2 August 2014

CHAPTER 1- Introduction to Differentiation

If $f(x)$ is a differentiable function for a given $x$, this means that we can draw a unique tangent to $f(x)$ for that given $x$. The slope of this unique tangent is called the derivative of $f(x)$ for that given $x$. The process of finding the derivative is known as differentiation.
For example, for $f\left( x \right) = {x^2}$, the derivative at any given $x$ has the value $2x$ (we evaluated this by first principles in the previous chapter.) This means that the slope of the tangent drawn to at any given $x$ has the numerical value $2x$.
Equivalently stated, we can differentiate $f\left( x \right) = {x^2}$ to get $f'\left( x \right) = 2x$.
{$f'(x)$ represents the derivative of $f(x)$}.
Recall that we can differentiate a function at a given point only if the $LHD$ and$RHD$ at that point have equal values. If they do, then
 $f'\left( x \right) = \dfrac{{d\left( {f\left( x \right)} \right)}}{{dx}} = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h} = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x - h} \right) - f\left( x \right)}}{{ - h}}$
The notation that we use to signify the derivative of $y = f\left( x \right)$ is either $f'\left( x \right)\left( {{\rm{or}}\,y'} \right)$ or $\dfrac{{d\,f\left( x \right)}}{{dx}}\left( {{\rm{or}}\dfrac{{dy}}{{dx}}} \right)$
We need to understand here the significance of the notation $\dfrac{{dy}}{{dx}}$ (the derivative of $y$ with respect to the variable $x$)
Recall that to evaluate the derivative (slope of the tangent) of $y=f(x)$ at a given point, we first drew a secant passing through that point and then let that secant tend to a tangent as follows:
The slope of any secant can be written easily:
The slope is obvious from the figure:
 $\tan \theta = \dfrac{{BC}}{{AB}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{\Delta y}}{{\Delta x}}$
Now, to evaluate the derivative at ${x_1}$, we need to make the secant $AC$ tend to a tangent at $A$ by letting ${x_2}$ approach ${x_1}\left( {{x_2} \to {x_1}} \right)$ or equivalently, by letting $\Delta x \to 0.$
As $\Delta x$ becomes an infinitesmally small quantity (approaches $0$), the corresponding $\Delta y$ will also become infinitesmally small (will approach $0$), but the ratio $\dfrac{{\Delta y}}{{\Delta x}}$ will become an increasingly accurate representation of the slope of the tangent at $A$.
An infinitesmally small change in the $x$ value is represented by $dx$ instead of $\Delta x$.
Similarly, an infinitesmally small change in the $y$ value would be represented by $dy$ instead of $\Delta y$.
Therefore,
 $\mathop {\lim }\limits_{{x_2} \to {x_1}} \dfrac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{\Delta y}}{{\Delta x}} = \dfrac{{dy}}{{dx}} = y'$
You should now be clear about the notation $\dfrac{{dy}}{{dx}}$. We will use $\dfrac{{dy}}{{dx}},\dfrac{{d\,f\left( x \right)}}{{dx}},\,f'\left( x \right)\,\,{\rm{or}}\,\,y'$ interchangeably to represent the derivative of $y = f\left( x \right)$ at any given $x$.
Note: From now on you should always keep in mind that $\Delta$ (variable) represents an infinitesmally small change in the variable value while (variable) represents a finite change in the variable value