Summary
Differentiation involves finding the derivative of a function, which represents the rate of change of the function's value with respect to a variable. It includes techniques like parametric differentiation, implicit differentiation, and understanding rates of change.
- Parametric Differentiation — Differentiating functions defined in terms of a third variable called a parameter. Example: If x and y are functions of t, differentiate using a variation of the chain rule.
- Implicit Differentiation — Differentiating functions that are not easily rearranged into y = f(x) or x = f(y). Example: Use the chain rule and product rule; results often involve both x and y.
- Rates of Change — Calculating how one variable changes with respect to another, often involving a third variable. Example: The rate of change of distance is velocity, calculated using derivatives.
Exam Tips
Key Definitions to Remember
- Parametric equations involve a third variable, the parameter.
- Implicit functions are not easily rearranged into y = f(x).
- Rate of change is the derivative of a function.
Common Confusions
- Mixing up parametric and implicit differentiation.
- Forgetting to apply the chain rule in implicit differentiation.
Typical Exam Questions
- What is the average velocity during a time interval? Calculate the difference in distance over the time interval and divide by the time.
- How do you find the gradient of a curve at a point? Differentiate the function and substitute the point's coordinates.
- How do you show a tangent is parallel to an axis? Set the derivative equal to zero for the x-axis or undefined for the y-axis.
What Examiners Usually Test
- Ability to differentiate parametric and implicit functions.
- Understanding of rates of change and their applications.
- Correct application of differentiation rules like the chain and product rules.