Parcijalne derivacije u SAGE-u

verzija: SageMath 9.4

In [1]:
%display latex
In [2]:
var('y z')
Out[2]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(y, z\right)\]

1. zadatak

Odredite parcijalne derivacije sljedećih funkcija:

  1. $g(x,y)=3x^2+xy+\sqrt{y}$
  2. $z=ye^y+\sqrt{x}$
  3. $u(x,y)=\frac{2x-y}{x+y}$
  4. $z=2^{\sin{\frac{y}{x}}}$
  5. $z=\ln{\left(\mathop{\mathrm{tg}}{\frac{x}{y}}\right)}$
  6. $f(x,y,z)=e^{2xz}-\log_2{(yz)}+1$

Rješenje

a) dio

In [3]:
diff(3*x^2+x*y+y^(1/2),x)
Out[3]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}6 \, x + y\]
In [4]:
diff(3*x^2+x*y+y^(1/2),y)
Out[4]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}x + \frac{1}{2 \, \sqrt{y}}\]

b) dio

In [5]:
diff(y*e^y+x^(1/2),x)
Out[5]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2 \, \sqrt{x}}\]
In [6]:
diff(y*e^y+x^(1/2),y)
Out[6]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}y e^{y} + e^{y}\]
In [7]:
factor(diff(y*e^y+x^(1/2),y))
Out[7]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}{\left(y + 1\right)} e^{y}\]
In [8]:
diff(y*e^y+x^(1/2),y).factor()
Out[8]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}{\left(y + 1\right)} e^{y}\]

c) dio

In [9]:
diff((2*x-y)/(x+y),x)
Out[9]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x - y}{{\left(x + y\right)}^{2}} + \frac{2}{x + y}\]
In [10]:
diff((2*x-y)/(x+y),x).factor()
Out[10]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, y}{{\left(x + y\right)}^{2}}\]
In [11]:
diff((2*x-y)/(x+y),y)
Out[11]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x - y}{{\left(x + y\right)}^{2}} - \frac{1}{x + y}\]
In [12]:
diff((2*x-y)/(x+y),y).factor()
Out[12]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{3 \, x}{{\left(x + y\right)}^{2}}\]

d) dio

In [13]:
diff(2^(sin(y/x)),x)
Out[13]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2^{\sin\left(\frac{y}{x}\right)} y \cos\left(\frac{y}{x}\right) \log\left(2\right)}{x^{2}}\]
In [14]:
diff(2^(sin(y/x)),y)
Out[14]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2^{\sin\left(\frac{y}{x}\right)} \cos\left(\frac{y}{x}\right) \log\left(2\right)}{x}\]

e) dio

In [15]:
diff(log(tan(x/y)),x)
Out[15]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\tan\left(\frac{x}{y}\right)^{2} + 1}{y \tan\left(\frac{x}{y}\right)}\]
In [16]:
diff(log(tan(x/y)),x).simplify_trig()
Out[16]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{y \cos\left(\frac{x}{y}\right) \sin\left(\frac{x}{y}\right)}\]
In [17]:
diff(log(tan(x/y)),y)
Out[17]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(\tan\left(\frac{x}{y}\right)^{2} + 1\right)} x}{y^{2} \tan\left(\frac{x}{y}\right)}\]
In [18]:
diff(log(tan(x/y)),y).simplify_trig()
Out[18]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{x}{y^{2} \cos\left(\frac{x}{y}\right) \sin\left(\frac{x}{y}\right)}\]

f) dio

In [19]:
diff(e^(2*x*z)-log(y*z,2)+1,x)
Out[19]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}2 \, z e^{\left(2 \, x z\right)}\]
In [20]:
diff(e^(2*x*z)-log(y*z,2)+1,y)
Out[20]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{y \log\left(2\right)}\]
In [21]:
diff(e^(2*x*z)-log(y*z,2)+1,z)
Out[21]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x e^{\left(2 \, x z\right)} - \frac{1}{z \log\left(2\right)}\]

2. zadatak

Odredite parcijalne derivacije drugog reda funkcije $z=\ln{\big(x^2+xy+y^2\big)}.$

Rješenje

In [22]:
f(x,y)=log(x^2+x*y+y^2)

Parcijalne derivacije prvog reda

In [23]:
diff(f(x,y),x)
Out[23]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, x + y}{x^{2} + x y + y^{2}}\]
In [24]:
diff(f(x,y),y)
Out[24]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x + 2 \, y}{x^{2} + x y + y^{2}}\]

Ako želimo parcijalne derivacije definirati kao nove funkcije

In [25]:
diff(f,x)
Out[25]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{2 \, x + y}{x^{2} + x y + y^{2}}\]
In [26]:
diff(f,y)
Out[26]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{x + 2 \, y}{x^{2} + x y + y^{2}}\]
In [27]:
fx(x,y)=diff(f,x)
fy(x,y)=diff(f,y)

Računanje vrijednosti parcijalnih derivacija u nekoj točki

In [28]:
fx(1,2)
Out[28]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4}{7}\]
In [29]:
fy(1,2)
Out[29]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{5}{7}\]

Gradijent funkcije

In [30]:
f(x,y).gradient()
Out[30]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{2 \, x + y}{x^{2} + x y + y^{2}},\,\frac{x + 2 \, y}{x^{2} + x y + y^{2}}\right)\]
In [31]:
f.gradient()
Out[31]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \left(\frac{2 \, x + y}{x^{2} + x y + y^{2}},\,\frac{x + 2 \, y}{x^{2} + x y + y^{2}}\right)\]
In [32]:
f.diff()
Out[32]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \left(\frac{2 \, x + y}{x^{2} + x y + y^{2}},\,\frac{x + 2 \, y}{x^{2} + x y + y^{2}}\right)\]
In [33]:
gradf=f.diff()

gradijent funkcije u točki $(1,2)$

In [34]:
gradf(1,2)
Out[34]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{4}{7},\,\frac{5}{7}\right)\]

Parcijalne derivacije drugog reda

In [35]:
diff(f(x,y),x,2)
Out[35]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(2 \, x + y\right)}^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{2}{x^{2} + x y + y^{2}}\]
In [36]:
diff(f(x,y),x,2).factor()
Out[36]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x^{2} + 2 \, x y - y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}}\]
In [37]:
diff(f(x,y),x,y)
Out[37]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(2 \, x + y\right)} {\left(x + 2 \, y\right)}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{1}{x^{2} + x y + y^{2}}\]
In [38]:
diff(f(x,y),x,y).factor()
Out[38]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{x^{2} + 4 \, x y + y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}}\]
In [39]:
diff(f(x,y),y,x)
Out[39]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(2 \, x + y\right)} {\left(x + 2 \, y\right)}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{1}{x^{2} + x y + y^{2}}\]
In [40]:
diff(f(x,y),y,x).factor()
Out[40]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{x^{2} + 4 \, x y + y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}}\]
In [41]:
diff(f(x,y),y,2)
Out[41]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(x + 2 \, y\right)}^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{2}{x^{2} + x y + y^{2}}\]
In [42]:
diff(f(x,y),y,2).factor()
Out[42]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x^{2} - 2 \, x y - 2 \, y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}}\]

Hesseova matrica

In [43]:
f.hessian()
Out[43]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \left( x, y \right) \ {\mapsto} \ -\frac{{\left(2 \, x + y\right)}^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{2}{x^{2} + x y + y^{2}} & \left( x, y \right) \ {\mapsto} \ -\frac{{\left(2 \, x + y\right)} {\left(x + 2 \, y\right)}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{1}{x^{2} + x y + y^{2}} \\ \left( x, y \right) \ {\mapsto} \ -\frac{{\left(2 \, x + y\right)} {\left(x + 2 \, y\right)}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{1}{x^{2} + x y + y^{2}} & \left( x, y \right) \ {\mapsto} \ -\frac{{\left(x + 2 \, y\right)}^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{2}{x^{2} + x y + y^{2}} \end{array}\right)\]
In [44]:
f.hessian().apply_map(lambda t:t.factor())
Out[44]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \left( x, y \right) \ {\mapsto} \ -\frac{2 \, x^{2} + 2 \, x y - y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} & \left( x, y \right) \ {\mapsto} \ -\frac{x^{2} + 4 \, x y + y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} \\ \left( x, y \right) \ {\mapsto} \ -\frac{x^{2} + 4 \, x y + y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} & \left( x, y \right) \ {\mapsto} \ \frac{x^{2} - 2 \, x y - 2 \, y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} \end{array}\right)\]
In [45]:
f.diff(2)
Out[45]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \left( x, y \right) \ {\mapsto} \ -\frac{{\left(2 \, x + y\right)}^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{2}{x^{2} + x y + y^{2}} & \left( x, y \right) \ {\mapsto} \ -\frac{{\left(2 \, x + y\right)} {\left(x + 2 \, y\right)}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{1}{x^{2} + x y + y^{2}} \\ \left( x, y \right) \ {\mapsto} \ -\frac{{\left(2 \, x + y\right)} {\left(x + 2 \, y\right)}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{1}{x^{2} + x y + y^{2}} & \left( x, y \right) \ {\mapsto} \ -\frac{{\left(x + 2 \, y\right)}^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} + \frac{2}{x^{2} + x y + y^{2}} \end{array}\right)\]
In [46]:
f.diff(2).apply_map(lambda t:t.factor())
Out[46]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \left( x, y \right) \ {\mapsto} \ -\frac{2 \, x^{2} + 2 \, x y - y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} & \left( x, y \right) \ {\mapsto} \ -\frac{x^{2} + 4 \, x y + y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} \\ \left( x, y \right) \ {\mapsto} \ -\frac{x^{2} + 4 \, x y + y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} & \left( x, y \right) \ {\mapsto} \ \frac{x^{2} - 2 \, x y - 2 \, y^{2}}{{\left(x^{2} + x y + y^{2}\right)}^{2}} \end{array}\right)\]
In [47]:
hessf=f.diff(2)

Hesseova matrica funkcije $f$ u točki $(1,2)$

In [48]:
hessf(x=1,y=2)
Out[48]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} -\frac{2}{49} & -\frac{13}{49} \\ -\frac{13}{49} & -\frac{11}{49} \end{array}\right)\]
In [49]:
hessf(1,2)
Out[49]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} -\frac{2}{49} & -\frac{13}{49} \\ -\frac{13}{49} & -\frac{11}{49} \end{array}\right)\]

3. zadatak

Pomoću diferencijala izračunajte približno $1.02^{2.99}.$

Rješenje

In [50]:
g(x,y)=x^y

gradijent funkcije $g$

In [51]:
grad_g=g.diff(); grad_g
Out[51]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \left(x^{y - 1} y,\,x^{y} \log\left(x\right)\right)\]

gradijent funkcije $g$ u točki $(1,3)$

In [52]:
grad_g(1,3)
Out[52]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(3,\,0\right)\]

Aproksimacija prirasta funkcije $g$ u točki $(1,3)$ pomoću diferencijala za pomake $\Delta x=0.02,\ \Delta y=-0.01$

In [53]:
v1=vector(grad_g(1,3))
v2=vector((0.02,-0.01))
v1.dot_product(v2)
Out[53]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}0.0600000000000000\]

"Egzaktna" vrijednost prirasta funkcije $g$ u točki $(1,3)$ za pomake $\Delta x=0.02,\ \Delta y=-0.01$

In [54]:
g(1.02,2.99)-g(1,3)
Out[54]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}0.0609978737408694\]

Aproksimacija vrijednosti $1.02^{2.99}$ pomoću diferencijala

In [55]:
g(1,3)+v1.dot_product(v2)
Out[55]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}1.06000000000000\]

"Egzaktna" vrijednost od $1.02^{2.99}$

In [56]:
g(1.02,2.99)
Out[56]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}1.06099787374087\]

4. zadatak

Zadana je funkcija $f(x,y)=x^2+y^2$. Odredite derivaciju funkcije $f$ duž vektora $\vec{u}=2\vec{i}+3\vec{j}$ u točki $(3,4)$.

Rješenje

In [57]:
f(x,y)=x^2+y^2
In [58]:
gradf=f.diff(); gradf
Out[58]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \left(2 \, x,\,2 \, y\right)\]

gradijent funkcije $f$ u točki $(3,4)$

In [59]:
gradf(3,4)
Out[59]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(6,\,8\right)\]

derivacija funkcije $f$ duž vektora $\vec{u}$ u točki $(3,4)$

In [60]:
gr=vector(gradf(3,4))
u=vector((2,3))
gr.dot_product(u)
Out[60]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}36\]

Parcijalne derivacije složenih funkcija

5. zadatak

Odredite $\frac{\mathrm{d}u}{\mathrm{d}t}$ ako je

$$u=\frac{z}{\sqrt{x^2+y^2}},\quad x=R\cos{t},\quad y=R\sin{t},\quad z=H.$$

Rješenje

In [61]:
var('R H t')
function('u');
In [62]:
u(x,y,z)=z/(sqrt(x^2+y^2))
In [63]:
assume(R>0)
u(R*cos(t),R*sin(t),H).simplify_full()
Out[63]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{H}{R}\]
In [64]:
diff(u(R*cos(t),R*sin(t),H),t)
Out[64]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}0\]

6. zadatak

Odredite $\frac{\partial z}{\partial u}$ i $\frac{\partial z}{\partial v}$ ako je

$$z=\mathop{\mathrm{arctg}}{\frac{x}{y}},\quad x=u\sin{v},\quad y=u\cos{v}.$$

Rješenje

In [65]:
function('x y z')
var('u v');
In [66]:
z(x,y)=arctan(x/y)
x(u,v)=u*sin(v)
y(u,v)=u*cos(v)
In [67]:
z(x(u,v),y(u,v))
Out[67]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\arctan\left(\frac{\sin\left(v\right)}{\cos\left(v\right)}\right)\]
In [68]:
z(x(u,v),y(u,v)).trig_reduce()
Out[68]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}v\]

$\mathop{\mathrm{arctg}}{\left(\mathop{\mathrm{tg}}{v}\right)}=v+k\pi,\ \ k\in\mathbb{Z}$

In [69]:
z(x(u,v),y(u,v)).trig_reduce()
Out[69]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}v\]
In [70]:
diff(z(x(u,v),y(u,v)),u)
Out[70]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}0\]
In [71]:
diff(z(x(u,v),y(u,v)),v)
Out[71]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}1\]

7. zadatak

Odredite $\frac{\partial z}{\partial x}$ i $\frac{\partial z}{\partial y}$ ako je

$$z=f(u,v),\quad u=x^2-y^2,\quad v=e^{xy}.$$

Rješenje

In [72]:
function('f u v')
var('x y')
Out[72]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(x, y\right)\]
In [73]:
u(x,y)=x^2-y^2
v(x,y)=e^(x*y)

$\frac{\partial z}{\partial x}=2xf_u(u,v)+ye^{xy}f_v(u,v)$

In [74]:
diff(f(u(x,y),v(x,y)),x)
Out[74]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}y e^{\left(x y\right)} \mathrm{D}_{1}\left(f\right)\left(x^{2} - y^{2}, e^{\left(x y\right)}\right) + 2 \, x \mathrm{D}_{0}\left(f\right)\left(x^{2} - y^{2}, e^{\left(x y\right)}\right)\]

$\frac{\partial z}{\partial y}=-2yf_u(u,v)+xe^{xy}f_v(u,v)$

In [75]:
diff(f(u(x,y),v(x,y)),y)
Out[75]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}x e^{\left(x y\right)} \mathrm{D}_{1}\left(f\right)\left(x^{2} - y^{2}, e^{\left(x y\right)}\right) - 2 \, y \mathrm{D}_{0}\left(f\right)\left(x^{2} - y^{2}, e^{\left(x y\right)}\right)\]

8. zadatak

Dokažite da funkcija $z=\varphi\big(x^2+y^2\big)$ zadovoljava parcijalnu diferencijalnu jednadžbu

$$y\frac{\partial z}{\partial x}-x\frac{\partial z}{\partial y}=0.$$

Rješenje

In [76]:
function('t z')
var('x y');
In [77]:
t(x,y)=x^2+y^2
In [78]:
zx=diff(z(t(x,y)),x); zx
Out[78]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x \mathrm{D}_{0}\left(z\right)\left(x^{2} + y^{2}\right)\]
In [79]:
zy=diff(z(t(x,y)),y); zy
Out[79]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}2 \, y \mathrm{D}_{0}\left(z\right)\left(x^{2} + y^{2}\right)\]
In [80]:
y*zx - x*zy
Out[80]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}0\]

Parcijalne derivacije implicitno zadane funkcije

In [81]:
var('x y z')
Out[81]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(x, y, z\right)\]

9. zadatak

Odredite $\frac{\mathrm{d}y}{\mathrm{d}x}$ i $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$ ako je

$$\big(x^2+y^2\big)^3-3\big(x^2+y^2\big)+1=0.$$

Rješenje

In [82]:
F(x,y)=(x^2+y^2)^3-3*(x^2+y^2)+1

$\dfrac{\mathrm{d}y}{\mathrm{d}x}$

In [83]:
F.implicit_derivative(y,x)
Out[83]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(x^{2} + y^{2}\right)}^{2} x - x}{{\left(x^{2} + y^{2}\right)}^{2} y - y}\]
In [84]:
F.implicit_derivative(y,x).simplify_full()
Out[84]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{x}{y}\]

$\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$

In [85]:
F.implicit_derivative(y,x,2)
Out[85]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{4 \, {\left(x^{2} + y^{2}\right)} x^{2} - \frac{4 \, {\left({\left(x^{2} + y^{2}\right)}^{2} x - x\right)} {\left(x^{2} + y^{2}\right)} x y}{{\left(x^{2} + y^{2}\right)}^{2} y - y} + {\left(x^{2} + y^{2}\right)}^{2} - 1}{{\left(x^{2} + y^{2}\right)}^{2} y - y} + \frac{{\left({\left(x^{2} + y^{2}\right)}^{2} x - x\right)} {\left(4 \, {\left(x^{2} + y^{2}\right)} x y - \frac{{\left({\left(x^{2} + y^{2}\right)}^{2} x - x\right)} {\left(4 \, {\left(x^{2} + y^{2}\right)} y^{2} + {\left(x^{2} + y^{2}\right)}^{2} - 1\right)}}{{\left(x^{2} + y^{2}\right)}^{2} y - y}\right)}}{{\left({\left(x^{2} + y^{2}\right)}^{2} y - y\right)}^{2}}\]
In [86]:
F.implicit_derivative(y,x,2).simplify_full()
Out[86]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{x^{2} + y^{2}}{y^{3}}\]

10. zadatak

Odredite $\frac{\partial z}{\partial x}$ i $\frac{\partial z}{\partial y}$ ako je

$$x^2-2y^2+3z^2-yz+y=0.$$

Rješenje

In [87]:
F(x,y,z)=x^2-2*y^2+3*z^2-y*z+y

$\dfrac{\partial z}{\partial x}$

In [88]:
F.implicit_derivative(z,x)
Out[88]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, x}{y - 6 \, z}\]

$\dfrac{\partial z}{\partial y}$

In [89]:
F.implicit_derivative(z,y)
Out[89]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{4 \, y + z - 1}{y - 6 \, z}\]

11. zadatak

Odredite $\frac{\partial z}{\partial x},\ \frac{\partial z}{\partial y},\ \frac{\partial x}{\partial y},\ \frac{\partial x}{\partial z},\ \frac{\partial y}{\partial x},\ \frac{\partial y}{\partial z}$ ako je

$$y\sin{x}+y^3\ln{z}+xyz^2=0.$$

Rješenje

In [90]:
F(x,y,z)=y*sin(x)+y^3*log(z)+x*y*z^2

$\dfrac{\partial z}{\partial x}$

In [91]:
F.implicit_derivative(z,x).simplify_full()
Out[91]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{z^{3} + z \cos\left(x\right)}{2 \, x z^{2} + y^{2}}\]

$\dfrac{\partial z}{\partial y}$

In [92]:
F.implicit_derivative(z,y).simplify_full()
Out[92]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{x z^{3} + 3 \, y^{2} z \log\left(z\right) + z \sin\left(x\right)}{2 \, x y z^{2} + y^{3}}\]

$\dfrac{\partial x}{\partial y}$

In [93]:
F.implicit_derivative(x,y).simplify_full()
Out[93]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{x z^{2} + 3 \, y^{2} \log\left(z\right) + \sin\left(x\right)}{y z^{2} + y \cos\left(x\right)}\]

$\dfrac{\partial x}{\partial z}$

In [94]:
F.implicit_derivative(x,z).simplify_full()
Out[94]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x z^{2} + y^{2}}{z^{3} + z \cos\left(x\right)}\]

$\dfrac{\partial y}{\partial x}$

In [95]:
F.implicit_derivative(y,x).simplify_full()
Out[95]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{y z^{2} + y \cos\left(x\right)}{x z^{2} + 3 \, y^{2} \log\left(z\right) + \sin\left(x\right)}\]

$\dfrac{\partial y}{\partial z}$

In [96]:
F.implicit_derivative(y,z).simplify_full()
Out[96]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x y z^{2} + y^{3}}{x z^{3} + 3 \, y^{2} z \log\left(z\right) + z \sin\left(x\right)}\]
In [ ]: