Common Derivative Application and Optimization Errors

Where Correct Derivatives Still Produce Wrong Answers

• Using the wrong independent variable. State what changes with respect to what. • Substituting a snapshot before differentiating a related-rate equation. Differentiate the general relationship first. • Treating positive acceleration as speeding up. Compare the signs of velocity and acceleration. • Calling every critical number an extremum. Verify with signs, values, or concavity. • Ignoring endpoints in closed-interval problems. Absolute extrema require endpoint evaluation. • Calling $f"=0$ an inflection point automatically. Concavity must actually change. • Optimizing an expression before reducing to one variable. Use the constraint. • Ignoring the feasible domain. Negative dimensions and impossible production levels are algebraic outputs, not valid designs. • Applying L'Hopital before checking the form. Substitute first. • Accepting a calculator or iteration without verification. Check the result in the original model and inspect convergence.