4 Differentiation of a general function from first principles

Consider the graph of $y=f\left(x\right)$ shown in Figure 7.

Figure 7 :

Carefully make the following observations:

1. Point $A$ has coordinates $\left(x,f\left(x\right)\right)$ .
2. Point $B$ has coordinates $\left(x+h,f\left(x+h\right)\right)$ .
3. The straight line $AB$ has gradient
   $$\frac{f\left(x+h\right)-f\left(x\right)}{h}$$
4. If we let $h\to 0$ we can find the gradient of the graph of $y=f\left(x\right)$ at the arbitrary

   point $A$ , provided we can evaluate the appropriate limit on $h$ . The resulting limit is the

   derivative of $f$ with respect to $x$ and is written $\frac{df}{dx}$ or $f^{\prime }\left(x\right)$ .

   Key Point 3

   Definition of Derivative

   Given $y=f\left(x\right)$ , its derivative is defined as

   $$\frac{df}{dx}=\frac{f\left(x+h\right)-f\left(x\right)}{h}\text{ in the limit as }h\text{ tends to 0.}$$

   This is written

   $$\frac{df}{dx}=\underset{h\to 0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}$$

   In a graphical context, the value of $\frac{df}{dx}$ at $A$ is equal to $tan\theta$ which is the tangent of the angle that the gradient line makes with the positive $x$ -axis.

Example 1

Differentiate $f\left(x\right)=x^2+2x+3$ from first principles.

Solution

$\frac{df}{dx}=\underset{h\to 0}{lim}\left\{\frac{f\left(x+h\right)-f\left(x\right)}{h}\right\}$

$=\underset{h\to 0}{lim}\left\{\frac{\left[{\left(x+h\right)}^2+2\left(x+h\right)+3\right]-\left[x^2+2x+3\right]}{h}\right\}$

$=\underset{h\to 0}{lim}\left\{\frac{\left[x^2+2xh+h^2+2x+2h+3-x^2-2x-3\right]}{h}\right\}$

$=\underset{h\to 0}{lim}\left\{\frac{2xh+h^2+2h}{h}\right\}$

$=\underset{h\to 0}{lim}\left\{2x+h+2\right\}$

$=2x+2$

Exercises

1. Use the definition of the derivative to find $\frac{df}{dx}$ when
   1. $f\left(x\right)=4x^2$ ,
   2. $f\left(x\right)=2x^3$ ,
   3. $f\left(x\right)=7x+3$ ,
   4. $f\left(x\right)=\frac{1}{x}$ . (Harder: try
   5. $f\left(x\right)=sinx$ and use the small angle approximation $sin\theta \approx \theta$ if $\theta$ is small and measured in radians.)
2. Using your results from Exercise 1 calculate the gradient of the following graphs at the given points:
   1. $f\left(x\right)=4x^2$ at $x=-2$ ,
   2. $f\left(x\right)=2x^3$ at $x=2$ ,
   3. $f\left(x\right)=7x+3$ at $x=-5$ ,

       

   4. $f\left(x\right)=\frac{1}{x}$ at $x=1∕2$ .
3. Find the rate of change of the function $y\left(x\right)=\frac{x}{x+3}$ at $x=3$ by considering the interval

     $x=3$ to $x=3+h$ .

Answer