2011-2012 Problem 11:
Chain Rule

Calculate the derivative of tan2(2x –1) with respect to x using the chain rule, and then verify your answer using a second differentiation technique. 

Solution:

Note that tan2(2x –1) = [tan (2x – 1)]2. To apply the chain rule, complete these steps:

1. Apply the power rule, changing the exponent of 2 into the coefficient of tan (2x – 1), and then subtracting 1 from the square.
2. Multiply by the expression tan (2x – 1), which was originally raised to the second power.
3. Take the derivative of tan (2x – 1) with respect to x.
4. Multiply by the derivative of 2x – 1, the expression that is plugged into tangent.

These four steps are implemented in the solution below. 

To verify the derivative, apply the product rule, noting that  tan2(2x –1) = tan (2x –1) · tan (2x – 1). Evev the product rule will require the chain rule, when you differentiate each factor (2x – 1), as demonstrated below. 

Both techniques result in the same derivative.

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