3 Derivative Of Simple Functions

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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 Formula, 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 




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 Formula.                                                          diff(2*x,x,1)




3. Consider f(x) = x2. Then Formula

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 Formula 



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


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




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

Figure 4: f(x) = Formula

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


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


More examples


1. Let f(x) = Formula. 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




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


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

(b) Formula                                                           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) = Formula                                                                                                                  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)


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