Consider solving the problem
\(y = \frac{x^2\sqrt{3x-2}}{(x+1)^2}\)
, find
\(\frac{dy}{dx}\)
# Need to define variables as symbolic for sympy to use them.
x
,
y
=
symbols
(
"x, y"
,
real
=
True
)
# Define y
y
=
x
**
2
*
sqrt
(
3
*
x
-
2
)
/
(
x
+
1
)
**
2
y
# This causes it to output formatted nicely. The print command formats it as computer code
\begin{equation*}
\frac{x^{2} \sqrt{3 x - 2}}{\left(x + 1\right)^{2}}
\end{equation*}
# Taking the natural log of this:
a
=
ln
(
x
**
2
*
sqrt
(
3
*
x
-
2
)
/
(
x
+
1
)
**
2
)
a
# Keep in mind that the left hand side is ln(y)
\begin{equation*}
\log{\left (\frac{x^{2} \sqrt{3 x - 2}}{\left(x + 1\right)^{2}} \right )}
\end{equation*}
# Taking its derivative with respect to x
# So, we took the log of the right side, then the derivative
diff
(
a
,
x
)
\begin{equation*}
\frac{\left(x + 1\right)^{2}}{x^{2} \sqrt{3 x - 2}} \left(\frac{3 x^{2}}{2 \left(x + 1\right)^{2} \sqrt{3 x - 2}} - \frac{2 x^{2} \sqrt{3 x - 2}}{\left(x + 1\right)^{3}} + \frac{2 x \sqrt{3 x - 2}}{\left(x + 1\right)^{2}}\right)
\end{equation*}
# That is kind of messy. Let the computer simplify it
simplify
(
diff
(
a
,
x
))
\begin{equation*}
\frac{3 x^{2} + 15 x - 8}{2 x \left(3 x^{2} + x - 2\right)}
\end{equation*}
# We could have done this all at once
simplify
(
diff
(
ln
(
y
),
x
))
\begin{equation*}
\frac{3 x^{2} + 15 x - 8}{2 x \left(3 x^{2} + x - 2\right)}
\end{equation*}
Now, the left hand side processed the same way is
\begin{equation*}
\end{equation*}
take the natural log
\begin{equation*}
\ln(y)
\end{equation*}
Take the derivative
\begin{equation*}
\frac{1}{y}dy
\end{equation*}
So, our entire processed equation is
\begin{equation*}
\frac{1}{y}dy = \frac{3 x^{2} + 15 x - 8}{2 x \left(3 x^{2} + x - 2\right)}
\end{equation*}
so all we have to do is multiply the processed right side by
\(y\)
and we have our answer and our derivative is
simplify
(
diff
(
ln
(
y
),
x
)
*
y
)
\begin{equation*}
\frac{x \left(3 x^{2} + 15 x - 8\right)}{2 \sqrt{3 x - 2} \left(x^{3} + 3 x^{2} + 3 x + 1\right)}
\end{equation*}
Of course, Sympy is capable enough that we can just take the derivative
directly
simplify
(
diff
(
y
,
x
))
\begin{equation*}
\frac{x \left(3 x^{2} + 15 x - 8\right)}{2 \sqrt{3 x - 2} \left(x^{3} + 3 x^{2} + 3 x + 1\right)}
\end{equation*}
but that was way too easy.
We could compare them, just to be sure they are the same
simplify
(
diff
(
y
,
x
))
==
simplify
(
diff
(
ln
(
y
),
x
)
*
y
)
Or plot this monstrosity
plot
((
diff
(
y
,
x
),(
x
,
0.2
,
10
)),
(
y
,
(
x
,
0.5
,
10
)))
# To change colors
# show = False delays the plot until we can set all of the parameters
# legend turns on the legend and uses the labels we have later.
p
=
plot
((
diff
(
y
,
x
),(
x
,
0.2
,
10
)),
(
y
,
(
x
,
0.5
,
10
)),
show
=
False
,
legend
=
True
)
p
[
0
]
.
line_color
=
'blue'
p
[
0
]
.
label
=
'$
\\
frac{dy}{dx}$'
p
[
1
]
.
line_color
=
'green'
p
[
1
]
.
label
=
'$y$'
p
.
show
()
