## 2011-2012 Problem 11:

Chain Rule

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

### Solution:

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

- Apply the power rule, changing the exponent of 2 into the coefficient of tan (2
*x*– 1), and then subtracting 1 from the square. - Multiply by the expression tan (2
*x*– 1), which was originally raised to the second power. - Take the derivative of tan (2
*x*– 1) with respect to*x.* - Multiply by the derivative of 2
*x*– 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(2*x* –1) = tan (2*x* –1) · tan (2*x* – 1). Evev the product rule will require the chain rule, when you differentiate each factor (2*x* – 1), as demonstrated below.

Both techniques result in the same derivative.