Differentiation

In this unit, students learn how to differentiate composite functions using the chain rule and apply that understanding to determine derivatives of implicit and inverse functions. Students need to understand that for composite functions, y is a function of u while u is a function of x. r. Mastering the chain rule is essential to success in all future units.

Identifying composite and implicit functions is a key differentiation skill. Students must recognize functions embedded in functions and be able to decompose composite functions into their “outer” and “inner” component functions. Misapplying the chain rule by forgetting to also differentiate the “inner” function or misidentifying the “inner” function are common errors.

The chain rule will be the target of assessment for many questions and a necessary step along the way for others. One common error is not recognizing when the chain rule applies, especially in composite functions.

The main question that drives this unit is:

1. If pressure experienced by a diver is a function of depth and depth is a function of time, how might we find the rate of change in pressure with respect to time?

• Differentiate a composite function
• Differentiate a power function
• Use the power rule for functions to find a derivative
• Use the chain rule for multiple composite functions
• Find a derivative using implicit differentiation
• Find higher order derivatives using implicit differentiation
• Differentiate functions with rational exponents
• Find the derivative of an inverse function
• Find the derivative of the inverse trig functions
• Differentiate logarithmic functions
• Express e as a limit

The last unit dealt with functions that were not composite functions. But, when you have a composite function and you have to take a derivative, then you must do chain rule. Usually, when you do chain rule, you also do u-substitution so that you can turn the function back into something that you have already learned.

There will be times when you will have to chain rule more than once. Pay attention to what you are taking the derivative of to make sure that you are not having to chain rule more than once.

All of the derivative rules are below. Each derivative rule is in terms of u so that you can see that you always need to do chain rule!

All the derivatives up to this point have been in just one variable. When you do implicit differentiation, you have more than one variable. The method used to differentiate an implicitly defined function is called implicit differentiation. If does not require rewriting the function explicitly, but it does require that the dependent variable is a differentiable function of the independent variable. Remember the following when you want to differentiate an implicit function.

1. Assume that y is a differentiable function of x

2. Differentiate BOTH sides of the equation with respect to x

3. Solve the resulting equation for y' or dy/dx

If a function has an inverse, then you can use the following formula to find the derivative of an inverse function.