Finding the Derivative where Relationships are Defined Parametrically

Finding the Derivative where Relationships are Defined Parametrically

Understanding Parametrically Defined Relationships

• Familiarise yourself with parametrically defined relationships - these define a relationship where both x and y are expressed as functions of a third variable, typically denoted as ‘t’.
• Understand that for these relationships, the independent variable is ‘t’ (the parameter), and the dependent variables are ‘x’ and ‘y’.
• For example, x = at^2 and y = bt^3 denote a parametrically defined relationship.

Derivatives in Parametric Equations

• Know the specific method to find the derivative of these relationships, which involves implicit differentiation.
• Recall that the derivative of y with respect to x in a parametric equation is given by the formula: dy/dx = (dy/dt) / (dx/dt)
• Understand that dy/dt is the derivative of y with respect to t, and dx/dt is the derivative of x with respect to t.

Performing Implicit Differentiation

• Practice performing differentiation with respect to ‘t’.
• For instance, to initiate the process of finding dy/dx, first find dy/dt and dx/dt.
• Then use the formula dy/dx = (dy/dt) / (dx/dt) to calculate the derivative of y with respect to x.

Exercises and Problem Solving

• Aim to become skilled in differentiation of standard functions with respect to ‘t’.
• Solve exercises that involve finding the derivatives of functions that are defined parametrically.
• Try to tackle progressively complex mathematical problems.
• See how these concepts are applied in solving real-world problems in areas like physics and geometry, where objects often move along parametrically defined paths.