Finding the Derivative where Relationships are Defined Implicitly

Finding the Derivative where Relationships are Defined Implicitly

Implicit Differentiation Basics

• Understand that implicit differentiation is a method used to find derivatives of equations where the dependent and independent variables can’t be separated easily.
• Familiarise yourself with the notation dy/dx and d/dx which signify differentiation with respect to x.
• Recognize that even though an equation may not be explicitly solved for y, you can still take the derivative with respect to x.

Performing Implicit Differentiation

• Know that when differentiating with respect to x, if you come across y, you must treat it like a function of x and apply the chain rule which includes multiplying by dy/dx.
• Understand how to differentiate each term of an equation with respect to x using your standard rules of differentiation (power rule, product rule, quotient rule, chain rule).

Applying Implicit Differentiation

• Be aware that implicit differentiation may result in an equation that includes dy/dx.
• Learn to isolate dy/dx to find the derivative of the unknown function.

Solving Problems with Implicit Differentiation

• Practice using implicit differentiation in problems with higher degrees (for example, square roots or trigonometric functions).
• Understand that dy/dx is itself a function of x. Substitute given values into this derivative function to find the rate of change at any given point.

Conceptual Understanding

• Grasp the idea that sometimes it’s impossible or impractical to solve an equation for y before differentiating. This is when implicit differentiation is particularly useful.
• Appreciate the application of implicit differentiation in solving complex mathematical problems where the relationship between variables is not a simple one.