Calcola con un opportuno script e/o R e/o WolframAlpha F'(2) e G'(π/2) dove F(x) = √(1+2x²) e G(x) = x²·sin(3x).

• Con R (vedi):
source("http://macosa.dima.unige.it/r.R")
f = function(x) sqrt(1+2*x^2); deriv(f,"x")
# 0.5 * (2 * (2 * x) * (1 + 2 * x^2)^-0.5)
# ossia: 2*x/sqrt(1+2*x^2)
df = function(x) eval(deriv(f,"x"))
df(2)              # 1.333333
fraction(df(2))    # 4/3
f = function(x) x^2*sin(3*x); deriv(f,"x")
# 2 * x * sin(3 * x) + x^2 * (cos(3 * x) * 3)
df = function(x) eval(deriv(f,"x")); df(pi/12); more(df(pi/12))
# 0.5156332   0.515633249005674
• Con WolframAlpha:
d sqrt(1+2*x^2) /dx
     2x
  ————————
  √(2x²+1)
d sqrt(1+2*x^2) /dx at x=2
  4/3
d x^2*sin(3*x) /dx
  x(3x cos(3x)+2 sin(3x))
d x^2*sin(3*x) /dx at x=pi/12
  π/12(√2+π/(4√2)) ≈ 0.515633249…

•  Utilizzando lo script "funct.(tab/limit)" (modificando opportunamente "F"):

    sqrt(1+2*pow(x,2))    x = 2 -> F(x) = 3
x1 = 1.99999, x2 = 2.00001 -> DF/Dx = 1.3333333333333333
x1 = 1.9999, x2 = 2.0001 -> DF/Dx = 1.33333333316754
x1 = 1.999, x2 = 2.001 -> DF/Dx = 1.3333333168722785
x1 = 1.99, x2 = 2.01 -> DF/Dx = 1.3333316872133374
x1 = 1.9, x2 = 2.1 -> DF/Dx = 1.3331684291609458

    pow(x,2)*sin(3*x)   x = 1.5707963267948966 -> F(x) = -2.4674011002723395
x1 = 1.5707963267948966, x2 = 1.57079632 -> DF/Dx = -3.141592694788039
x1 = 1.5707963267948966, x2 = 1.5707963 -> DF/Dx = -3.1415929262207607
x1 = 1.5707963267948966, x2 = 1.570796 -> DF/Dx = -3.141595954649396
x1 = 1.5707963267948966, x2 = 1.57079 -> DF/Dx = -3.1416565745920773
x1 = 1.5707963267948966, x2 = 1.5707 -> DF/Dx = -3.142565741390576
    [ π = 3.141592653589793 ]