#Calculate the first derivative of the following functions:
# 2*x^4 + 3*x^2 + 2*x + 7
f(x) = 3*x^6 - 2*x^4 + 3*x^2 + 2*x + 7
diff( f(x), x)

#Graph the second and fourth derivative from the function above
f(x) = 3*x^6 - 2*x^4 + 3*x^2 + 2*x + 7
diff(f(x), x, 2)
diff(f(x), x, 4)

#Graph 2*sin(2*x) and its derivative on a graph, using two different colors to represent the different functions

f(x) = 2*sin(2*x)
g(x) = diff(f(x), x)

plot(f(x), -pi, pi) + plot (g(x), -pi, pi, color='red')

#Find the point of intersections between 0 and pi

f(x) = 2*sin(2*x)
g(x) = diff(f(x), x)

find_root( f(x) == g(x), 0, pi/2)

find_root( f(x) == g(x), pi/2, pi)

#Graph the points of intersection
a= 0.5535743588970452
b= 2.1243706856922784

f(x) = 2*sin(2*x)
g(x) = diff(f(x), x)

f(a)

a= 0.5535743588970452
b= 2.1243706856922784

f(x) = 2*sin(2*x)
g(x) = diff(f(x), x)


plot( 2*sin(2*x), -pi, pi) + plot (g(x), -pi, pi, color= 'red') + point ((2.1243706856922784, -1.78885438199983), size= 50 ) + point ((0.5535743588970452, 1.78885438199983), size = 50)