9–61. Evaluate and simplify y'.


y = x²+2x tan^−1(cot x)

Verified step by step guidance

1. First, identify the function y given in the problem. Here, y = x² + 2x tan^−1(cot x). We need to find the derivative y'.
2. Apply the derivative rules to each term separately. Start with the derivative of x², which is straightforward using the power rule: d/dx(x²) = 2x.
3. Next, find the derivative of 2x tan^−1(cot x). Use the product rule for derivatives, which states that d/dx[u*v] = u'v + uv'. Here, u = 2x and v = tan^−1(cot x).
4. Calculate the derivative of u = 2x, which is simply 2. Then, find the derivative of v = tan^−1(cot x). This requires using the chain rule: d/dx[tan^−1(u)] = 1/(1+u²) * du/dx, where u = cot x.
5. The derivative of cot x is -csc²x. Substitute this into the chain rule expression to find the derivative of tan^−1(cot x). Combine all these derivatives using the product rule to find y'.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the given function y, which includes polynomial and inverse trigonometric components.

Chain Rule

The Chain Rule is a technique used in differentiation when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This rule is essential for differentiating the term involving tan^−1(cot x) in the given function.

Product Rule

The Product Rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions u(x) and v(x), the derivative of their product is u'v + uv'. In the given function, if any terms are products of functions, this rule will be necessary to apply during differentiation.