CalculusEasyMade_CH5

5_2

$y=\sqrt{x^2+a^2}$

a, x, u=symbols('a x, u', real=True)
y=sqrt(x**2+a**2)
y
$\sqrt{a^{2} + x^{2}}$

u1=x**2+a**2
dudx=diff(u1, x)
dudx
2x

y1=sqrt(u)
dy1du=diff(y1, u)
dy1du
$\frac{1}{2\sqrt{u}$

dy1dx=dy1du*dudx
dy1dx.subs(u, u1)
$\frac{x}{\sqrt{a^{2} + x^{2}}}$

dydx=diff(y, x)
dydx
$\frac{x}{\sqrt{a^{2} + x^{2}}}$

5_6

$\begin{align}y&=\frac{(x^4+a)^{\frac{3}{2}}}{x^3+a}\\ &=\frac{u}{v}\\ u&=x^4+a\\ v&=x^3+a\\ dy&=\frac{du^{\frac{3}{2}} \cdot v-u^{\frac{3}{2}} \cdot dv}{v^2}\end{align}$

diff() 함수를 사용하여 직접 미분한 결과는 다음과 같습니다.

a, x, u, v=symbols('a x, u, v', real=True)
y=(x**4+a)**Rational('3/2')/(x**3+a)
y
$\frac{\left(a + x^{4}\right)^{\frac{3}{2}}}{a + x^{3}}$

dydx=diff(y, x)
dydx
$\frac{6 x^{3} \sqrt{a + x^{4}}}{a + x^{3}} - \frac{3 x^{2} \left(a + x^{4}\right)^{\frac{3}{2}}}{\left(a + x^{3}\right)^{2}}$

위에서 전개된 방식대로 u, v에 치환에 의한 결과는 다음과 같습니다.

u1=x**4+a
dudx=diff(u1, x)
y1=u**Rational('3/2')
dy1du=diff(y1, u)
dy1dx=dy1du*dudx
dy1dx
$6 \sqrt{u} x^{3}$

dy1dx=dy1dx.subs(u, u1)
dy1dx
$6 x^{3} \sqrt{a + x^{4}}$

v1=x**3+a
dvdx=diff(v1, x)
y2=v
dy2dv=diff(y2,v)
dy2dx=dy2dv*dvdx
dy2dx
3x²

dy2dx=dy2dx.subs(v, v1)
dy2dx
$- \frac{3 x^{2}}{\left(a + x^{3}\right)^{2}}$

(dy1dx*(v1)-u1**Rational('3/2')*dy2dx)/v1**2
$\frac{6 x^{3} \left(a + x^{3}\right) \sqrt{a + x^{4}} - 3 x^{2} \left(a + x^{4}\right)^{\frac{3}{2}}}{\left(a + x^{3}\right)^{2}}$

5_10

$\begin{align}y&=\frac{1}{2}x^3\\ v&=u+u^2\\ w&=\frac{1}{v^2}\\ \frac{dw}{dx}&=\frac{dw}{dv}\cdot \frac{dv}{du} \cdot \frac{du}{dx}\end{align}$

x, u, v, w=symbols('x, u, v, w', real=True)
u1=1/2*x**3
v1=u+u**2
w=1/v**2
dwdv=diff(w, v)
dwdv
$-\frac{2}{v^3}$

dvdu=diff(v1, u)
dvdu
2u+1

dudx=diff(u1, x)
dudx
1.5x²

dwdx=dwdv*dvdu*dudx
dwdx
$- \frac{3.0 x^{2} \left(2 u + 1\right)}{v^{3}}$

dwdx=dwdx.subs({u:u1, v:v1})
dwdx
$- \frac{3.0 x^{2} \left(1.0 x^{3} + 1\right)}{\left(u^{2} + u\right)^{3}}$

dwdx=dwdx.subs(u, u1)
dwdx
$- \frac{24.0 x^{2} \left(1.0 x^{3} + 1\right)}{\left(0.5 x^{6} + x^{3}\right)^{3}}$

함수 w를 x에 관해 정리한 후 미분한 결과입니다.

w2=w.subs(v, v1).subs(u, u1)
w2
$\frac{4.0}{\left(0.5 x^{6} + x^{3}\right)^{2}}$

dw2dx=diff(w2, x)
dw2dx
$\frac{4.0 \left(- 6.0 x^{5} - 6 x^{2}\right)}{\left(0.5 x^{6} + x^{3}\right)^{3}}$