The derivative of a combined function

Given the function :
Determine the derivative in $val20

=

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

The derivative of a polynomial

Let be a polynomial, defined in $m_RR by .

Let be differentiable in $m_RR. Determine the derivative.

For all real ,   =

The product rule

Determine the derivative of a function in $m_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 $m_RR by   .

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

Tangent and derivative

Given the plane .

The curve $m_C is the graph of the function , defined in $val16.

The line is the tangent of $m_C in point , with coordinates ($val13 : $val24).

Point , with coordinates ($val28 : $val31) is also on line . Determine the value of     at two decimals accurate.

xrange -$val7,$val7 yrange -$val8,$val8 parallel -$val7,-$val8,-$val7,$val8,1,0, 2*$val7+1, grey parallel -$val7,-$val8,$val7,-$val8,0,1, 2*$val8+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 $val32 $val33 text blue , -$val7+0.5 ,$val25 , medium, y=f(x) $val39 linewidth 1.5 plot blue, $val15 plot green, $val23

Investigate a Function

Given the function defined in $m_RR by   .

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

  1. The function is differentiable in $m_RR :
    for all real :  
  2. The nature of the derivative function :
    The derivative of is zero for =
  1. The derivative of :
  2. The nature of the sign of :
    -$val17 $val15 +$val17
    0
  3. The nature of the sign of :
    • is on the interval $val18
    • is on the interval $val19
  4. The extremum of :
    • reaches in a with value