2.8 Embedded Sagemath

# Define a symbolic variable
x = var('x')

# Define a function
f = sin(x) * e^(-x/5)

# Take the derivative
f_derivative = diff(f, x)

# Solve for critical points (where derivative is zero)
critical_points = solve(f_derivative == 0, x)

# Display the function and its derivative
show(f)
show(f_derivative)
show(critical_points)

# Plot the function and its derivative
plot1 = plot(f, (x, 0, 10), color='blue', legend_label='f(x)')
plot2 = plot(f_derivative, (x, 0, 10), color='red', linestyle='--', legend_label="f'(x)")
combined_plot = plot1 + plot2
combined_plot.show()

$$/*<![CDATA[*/\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(-\frac{1}{5} \, x\right)} \sin\left(x\right)/*]]>*/$$