#1) Don't forget the Product Rule!
f(x)=x^2+1
g(x)=x^3+3
show(diff(f(x),x))
show(diff(g(x),x))
show(diff(f(x)*g(x),x).expand())
show((f(x)*diff(g(x),x)+g(x)*diff(f(x),x)).expand())
p1=plot(f(x)*g(x),0,5,color='red')
p2=plot(diff(f(x)*g(x),x),0,5,color='green')
p1+p2
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#2) Now try the Qutient Rule!
f(x)=x^2-1
g(x)=x^2+1
show(diff(f(x),x))
show(diff(g(x),x))
show(diff(f(x)/g(x),x).simplify_rational())
p1=plot(f(x)/g(x),-5,5,color='red')
p2=plot(diff(f(x)/g(x),x),-5,5,color='green')
p1+p2
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#3) find the tangent line to f(x)=x+2/x when x=1
f(x)=x+2/x
g(x)=diff(f(x),x)
show(f(x))
show(g(x))
show(f(1))
show(g(1))
p1=plot(f(x),0.5,3,color='red')
p2=plot(g(1)*(x-1)+f(1),0.5,3,color='green')
(p1+p2).show(aspect_ratio=1)
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