Finding Derivatives in Python

Suppose a mass \(m\) is dragged across the ground by a rope attached to a pulley. Suppose that \(x\) is the horizontal distance from the mass to the pulley, \(h\) is the height of the pulley, and \(F\) is the force applied to the rope. Let \(g\) be the acceleration due to gravity, and \(\mu\) be the friction coefficient of the box along the ground. It can be shown that the force is given by this equation
\[F=\frac{\mu m g\sqrt{x^2+h^2}}{x+\mu h}\]
How fast is the force changing with respect to \(x\) in general? How fast is it changing when \(m=18,\) \(h=10,\) \(\mu=0.55\), \(g=9.8,\) \(x=10\) and \(x=40.\)

Reveal Answer

Python Code:
from sympy import *

mu,m,g,x,h=symbols('mu m g x h')
F=mu*m*g*sqrt(x**2+h**2)/(x+mu*h)
dF=F.diff(x)print('The general derivative is',dF)
dFat10=F.subs({m:18,h:10,mu:0.55,g:9.8,x:10})
print('When x=10, the rate of change in t is',dFat10.evalf() ,'Newtons/meter')

dFat40=F.subs({m:18,h:10,mu:0.55,g:9.8,x:40})
print('When x=40, the rate of change in t is', dFat40.evalf(), 'Newtons/meter')

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