OEF derivative --- Introduction ---

In this module there are 6 exercices on differentiation .

The derivative of a combined function

Given the function :
Determine the derivative in

=

NB : write "sqrt(a x+b)" for

The derivative of a polynomial

Let be a polynomial, defined in RR by .

Let be differentiable in RR. Determine the derivative.

For all real ,   =

The product rule

Determine the derivative of a function in RR defined by with :

The functions and are differentiable in and :

=
=

In order to determine the derivative of   we apply the following rule of differentiation:

The derivative function of   will be :

=

The quotient rule

Given the function defined in RR by   .

We will now determine the derivative of    in a few steps :

Tangent and derivative

Given the plane .

The curve C is the graph of the function , defined in .

The line is the tangent of C in point , with coordinates ( : ).

Point , with coordinates ( : ) is also on line . Determine the value of     at two decimals accurate.

xrange -, yrange -, parallel -,-,-,,1,0, 2*+1, grey parallel -,-,,-,0,1, 2*+1, grey hline 0,0,black vline 0,0,black arrow 0,0,1,0,8, black arrow 0,0,0,1,8, black text black , -0.5,-0.3,small , O text black , 1,-0.3,small , I text black , -0.5,1,small , J text blue , -+0.5 , , medium, y=f(x) linewidth 1.5 plot blue, plot green,

Investigate a Function

Given the function defined in RR by   .

Investigate this function and determine the extremum (extrema) of .

  1. The function is differentiable in RR :
    for all real
  2. The nature of the derivative function is zero for =
  1. The derivative of
  2. The nature of the sign of - + 0
  3. The nature of the sign of is on the interval
  4. is on the interval
  5. The extremum of reaches in a with value
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