Derivative of Simple Functions

Bridge Course, August 2012

 

Intuition: The derivative of a function f(x) assigns measures the “slope” (i.e. how quickly the value of f(x) changes as the value of x changes) of the function at the point x. It is denoted , or just f′(x).

Figure 1: f(x) = e^x with a tangent to f at x = 2.5

 

Definition: Consider a function f = f(x) defined over real-line. Then f′(x) is defined as 

                                               .

 

Note:

The Definition can also looks like

for putting epsilon equal to negative of the previous defined one.

 

And a more fundamental definition which consider the derivative of a function at particular point c

From these definitions,

we know that if derivative of a function exists , then 

is also the derivative of f.

(Note : here we assumed we know that sum of limits is also a limit)

 

Examples: Derivative computation

Consider a few examples of derivative computation:

 

When limits exist                                                                          Maxima CODE

1. Consider f(x) = x. Then                                                                                                          diff(x,x,1)

                                                    

    Hence the slope of the function is 1 everywhere.

 

 

 

2. Consider f(x) = 2x. Then .                                                          diff(2*x,x,1)

 

 

 

3. Consider f(x) = x². Then . 

Hence the slope of the function changes with x – in particular, the larger the magnitude of x, the larger the slope.                                                                                                                                                  diff(x∧2,x,1)

 

  

                     (a) f'(0)                                          (b) f'(1)                                         (c) f'(2)  

Figure 2: slope of the function changes with different x

 

When derivatives do not exist

 

1. |x|, for x = 0.                                                                                                                diff(abs(x),x,1)

Figure 3: f(x) = |x|

Proof: we have

                                                                 

Assume the derivative exists.

Consider the sequence  

Then,

  

Note that: If n is odd, then  is -1.If n is even, then  is 1.

 

Therefore,  does not exist and hence derivative of |x|, for x=0 does not exist.

 

 

 

2.  for x = 0.                                                                                                                diff(1/x,x,1)

Figure 4: f(x) = 

Proof: Consider the function f(x) =  for x ̸= 0. We have

                         

Note that: If a is equal 0, the derivative of f(x) becomes negative infinity. Therefore, the derivative of  for x = 0 does not exist.

 

More examples

 

1. Let f(x) = . Find f′(x)                                                                                           diff(x∧n,x,1)

                         

2. Let f(x) = sin x. Find f′(x)                                                                                          diff(sin(x),x,1)

                         

Another Intuitive way :

3. Let f(x) = ln x. Find f′(x) and f′(100)                                                                           diff(log(x),x,1)

                         

And hence, f′(100) = 1/100

 

Detailed derivation for the above example

 

Useful limits before doing exercise:

Link to the derivation to the above limits

 

Exercises

 

1. Compute the derivative of f(x) by using definition of derivative

 

(a) f(x) =  + 2x + 1                                                                                         diff(x∧3+2*x+1,x,1)

(b)                                                            diff(x∧2,x,1) and diff(-x∧2,x,1)

(c) f(x) = cos x                                                                                                              diff(cos(x),x,1)

(d) f(x) = tan x                                                                                                              diff(tan(x),x,1)

(e) f(x) =                                                                                                                   diff(e∧x,x,1)

 

2. Determine if f′(x) exists at x = 0 and find the value if it exists

 

(a) f(x) = ln x                                                                                                                 diff(log(x),x,1)

(b) f(x) = ln(1 + x)                                                                                                      diff(log(1+x),x,1)