Carefully identify what quantity is changing with respect to time (related rates) versus what quantity needs to be maximized or minimized (optimization). Draw diagrams and label variables clearly.
Forgetting to differentiate all relevant variables with respect to time (dt). Ensure every term that depends on time also gets a dt component after differentiation.
Practice a wide variety of problems and pay close attention to how each variable is treated when differentiating with respect to another variable.
Defining the objective function accurately (the function to be maximized or minimized) and expressing it in terms of a single variable using constraint equations.
Make sure your calculator is in radian mode, and remember the derivatives of trigonometric functions (including the chain rule when applicable).
Failing to realize that the shortest distance is along the normal to the curve.
Always relate your answer back to the original problem context. Does your answer make sense within the given scenario? Include units where appropriate.
Break down the problem into smaller, manageable steps. Draw diagrams, define variables, write down relevant formulas, and check your work carefully.