A ratio changes when either its numerator or its denominator changes.
WHAT HAPPENS IF…At x=1, what is the derivative of (x²+1)/(x+2)?
Interactive exploration
Loading the mathematical model…
Connect the picture to the idea(gf)′=g2f′g−fg′,g=0
Write f=qg and differentiate with the product rule: f′=q′g+qg′. Solving for q′ gives the quotient rule. For f=x²+1 and g=x+2, the derivative is (x²+4x−1)/(x+2)². This experiment stays away from x=−2.
A detail that matters
The derivative of a quotient is not the quotient of the derivatives.