Correction de la fiche d'exercices n° 1 sur la dérivation

In [10]:
from sympy import *
init_session()
init_printing(pretty_print = True, use_unicode = True)

IPython console for SymPy 0.7.5 (Python 3.4.3-32-bit) (ground types: python)

These commands were executed:
>>> from __future__ import division
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)

Documentation can be found at http://www.sympy.org

Exercice 5

In [58]:
def deriver(exp):
    return diff(exp, x)
In [56]:
f1 = (5-2*x)/3
f1
Out[56]:
$$- \frac{2 x}{3} + \frac{5}{3}$$
In [28]:
deriver(f1)
Out[28]:
$$- \frac{2}{3}$$
In [30]:
f2 = x**6 - 4*x**5 + 3*x**3 - x + 7
In [31]:
deriver(f2)
Out[31]:
$$6 x^{5} - 20 x^{4} + 9 x^{2} - 1$$
In [41]:
f3 = x**3 - x**2 + 5 - 4*x
In [43]:
deriver(f3)
Out[43]:
$$3 x^{2} - 2 x - 4$$
In [44]:
f4 = x**3/2 - 3*x**2 + 5/x
In [45]:
f4
Out[45]:
$$\frac{x^{3}}{2} - 3 x^{2} + \frac{5}{x}$$
In [46]:
deriver(f4)
Out[46]:
$$\frac{3 x^{2}}{2} - 6 x - \frac{5}{x^{2}}$$
In [47]:
f5 = (1-2*x)*(0.5*x**2 + 1)
In [48]:
f5
Out[48]:
$$\left(- 2 x + 1\right) \left(0.5 x^{2} + 1\right)$$
In [49]:
deriver(f5)
Out[49]:
$$- 1.0 x^{2} + 1.0 x \left(- 2 x + 1\right) - 2$$
In [50]:
f6 = 1/x - x**4 + 3*x**2 + 5/x
In [51]:
deriver(f6)
Out[51]:
$$- 4 x^{3} + 6 x - \frac{6}{x^{2}}$$
In [52]:
f7 = (3*x**2 + 1)*(2*x-3)
In [53]:
f7
Out[53]:
$$\left(2 x - 3\right) \left(3 x^{2} + 1\right)$$
In [60]:
deriver(f7)
Out[60]:
$$6 x^{2} + 6 x \left(2 x - 3\right) + 2$$
In [61]:
f8 = (3*x**2 + x**4 - 5**3 + 1)**2
In [62]:
f8
Out[62]:
$$\left(x^{4} + 3 x^{2} - 124\right)^{2}$$
In [63]:
deriver(f8)
Out[63]:
$$\left(8 x^{3} + 12 x\right) \left(x^{4} + 3 x^{2} - 124\right)$$
In [64]:
f9 = 6*sqrt(x) + 3/x - 2/x**5
In [65]:
f9
Out[65]:
$$6 \sqrt{x} + \frac{3}{x} - \frac{2}{x^{5}}$$
In [66]:
deriver(f9)
Out[66]:
$$- \frac{3}{x^{2}} + \frac{10}{x^{6}} + \frac{3}{\sqrt{x}}$$
In [67]:
f10 = 6*x*sqrt(x)
In [68]:
f10
Out[68]:
$$6 x^{\frac{3}{2}}$$
In [69]:
deriver(f10)
Out[69]:
$$9 \sqrt{x}$$
In [70]:
f11 = (7-2*x)*sqrt(x)
In [71]:
f11
Out[71]:
$$\sqrt{x} \left(- 2 x + 7\right)$$
In [72]:
deriver(f11)
Out[72]:
$$- 2 \sqrt{x} + \frac{1}{2 \sqrt{x}} \left(- 2 x + 7\right)$$
In [73]:
f12 = (5*x-3)/(x-2)
In [74]:
f12
Out[74]:
$$\frac{5 x - 3}{x - 2}$$
In [75]:
deriver(f12)
Out[75]:
$$\frac{5}{x - 2} - \frac{5 x - 3}{\left(x - 2\right)^{2}}$$
In [76]:
f13 = 2+5/(x-1)
In [77]:
f13
Out[77]:
$$2 + \frac{5}{x - 1}$$
In [78]:
deriver(f13)
Out[78]:
$$- \frac{5}{\left(x - 1\right)^{2}}$$
In [79]:
f14 = 3 - 2*x-5*x/(x+1)
In [80]:
f14
Out[80]:
$$- 2 x - \frac{5 x}{x + 1} + 3$$
In [81]:
deriver(f14)
Out[81]:
$$\frac{5 x}{\left(x + 1\right)^{2}} - 2 - \frac{5}{x + 1}$$
In [82]:
factor(_)
Out[82]:
$$- \frac{1}{\left(x + 1\right)^{2}} \left(2 x^{2} + 4 x + 7\right)$$
In [83]:
f15 = sqrt(x)/(x-1)
In [84]:
f15
Out[84]:
$$\frac{\sqrt{x}}{x - 1}$$
In [85]:
deriver(f15)
Out[85]:
$$- \frac{\sqrt{x}}{\left(x - 1\right)^{2}} + \frac{1}{2 \sqrt{x} \left(x - 1\right)}$$
In [86]:
factor(_)
Out[86]:
$$- \frac{x + 1}{2 \sqrt{x} \left(x - 1\right)^{2}}$$
In [88]:
f16 = (x**2 + 1)/(x**2 + 2)
In [89]:
f16
Out[89]:
$$\frac{x^{2} + 1}{x^{2} + 2}$$
In [90]:
deriver(f16)
Out[90]:
$$- \frac{2 x \left(x^{2} + 1\right)}{\left(x^{2} + 2\right)^{2}} + \frac{2 x}{x^{2} + 2}$$
In [91]:
factor(_)
Out[91]:
$$\frac{2 x}{\left(x^{2} + 2\right)^{2}}$$
In [93]:
f17 = 1/(7-5*x)
In [94]:
f17
Out[94]:
$$\frac{1}{- 5 x + 7}$$
In [95]:
factor(deriver(f17))
Out[95]:
$$\frac{5}{\left(5 x - 7\right)^{2}}$$
In [96]:
f18 = 5/(x**2 - x)
In [97]:
f18
Out[97]:
$$\frac{5}{x^{2} - x}$$
In [98]:
factor(deriver(f18))
Out[98]:
$$- \frac{10 x - 5}{x^{2} \left(x - 1\right)^{2}}$$
In [99]:
f19 = (x**3 + 1)/(x**2 - 2)
In [100]:
factor(deriver(f19))
Out[100]:
$$\frac{x}{\left(x^{2} - 2\right)^{2}} \left(x^{3} - 6 x - 2\right)$$