Question

Extended Product Rule Derive a formula for the derivative of the product \(f g h\) of three differentiable functions.

Short answer

The derivative of the product of functions \( f \), \( g \), and \( h \) is given by the expression: \( (fgh)' = f'gh + fgh' + fg'h' \)

Step-by-step solution

Identify the Functions

From the exercise, we have three differentiable functions which we will denote as \( f \), \( g \), and \( h \). Our task is to determine the derivative of the product of these three functions.

Apply Extended Product Rule

The product rule for two functions says that \((fg)' = f'g + fg'\). To extend this to three functions, we treat one of the functions, say \( h \), as a constant and differentiate the product of the other two, followed by including the derivative of \( h \) and alternating this process. This would give us:\((fgh)' = (fg)'h + fg'h' = (f'g + fg')h + fg'h'\)

Expand and Simplify

Expand the above equation and group similar terms:\((fgh)' = f'gh + fgh' + fg'h'\)

Final Expression

The final expression for the derivative of the product of three functions is therefore: \((fgh)' = f'gh + fgh' + fg'h'\)