2011-2012 Problem 11:
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.
Note that tan2(2x –1) = [tan (2x – 1)]2. To apply the chain rule, complete these steps:
- Apply the power rule, changing the exponent of 2 into the coefficient of tan (2x – 1), and then subtracting 1 from the square.
- Multiply by the expression tan (2x – 1), which was originally raised to the second power.
- Take the derivative of tan (2x – 1) with respect to x.
- 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.