Problem 59 - Single Variable Calculus

Question

Find the derivative of the function. $$ g(x)=\tan ^{-1} x+x \cot ^{-1} x $$

Step-by-Step Solution

Answer

The short version of the solution is: To find the derivative of the given function, \(g'(x) = \frac{d}{dx}(\tan^{-1}(x)) + \frac{d}{dx}(x\cot^{-1}(x))\), we first recall the derivatives of arctangent and arc cotangent functions. Then, applying the product rule and sum rule, we obtain the derivative: $$ g'(x) = \frac{1-x}{1+x^2} + \cot^{-1}(x) $$

Step 1: Write down the given function and differentiate with respect to x

Given function: $$ g(x) = \tan^{-1}(x) + x\cot^{-1}(x) $$ Now, differentiate \(g(x)\) with respect to x: $$ g'(x) = \frac{d}{dx}(\tan^{-1}(x)) + \frac{d}{dx}(x\cot^{-1}(x)) $$

Step 2: Differentiate the arctangent function

We recall that the derivative of arctangent is given by: $$ \frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2} $$

Step 3: Apply the product rule to differentiate the second term

To differentiate \(x\cot^{-1}(x)\), we apply the product rule since it's a product of two functions: \(x\) and \(\cot^{-1}(x)\). The product rule states: $$ (uv)' = u'v + uv' $$ Here, \(u = x\) and \(v = \cot^{-1}(x)\). Differentiate \(x\) with respect to \(x\): $$ \frac{d}{dx}(x) = 1 $$ We recall that the derivative of arc cotangent is given by: $$ \frac{d}{dx}(\cot^{-1}(x)) = \frac{-1}{1+x^2} $$

Step 4: Apply the product rule and simplify

Now, substitute the derivatives back into the product rule: $$ (1)\left(\cot^{-1}(x)\right) + (x)\left(\frac{-1}{1+x^2}\right) $$ Simplify the expression: $$ \cot^{-1}(x) - \frac{x}{1+x^2} $$

Step 5: Combine the derivatives

Finally, combining the derivative of the arctangent and the result from the product rule, we get: $$ g'(x) = \frac{1}{1+x^2} + \cot^{-1}(x) - \frac{x}{1+x^2} $$ Thus, the derivative of the given function is: $$ g'(x) = \frac{1-x}{1+x^2} + \cot^{-1}(x) $$

Key Concepts

Arctangent Function

The arctangent function, denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \), is the inverse of the tangent function. It is used to retrieve the angle whose tangent is \( x \). This function is continuous for all real numbers and its range is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
When differentiating the arctangent function, it's important to remember its derivative formula:

• \( \frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2} \)

This simple formula results from the relationship between the derivative of the inverse trigonometric functions and their respective identities. The beauty of this particular derivative is in its simplicity, where the only variable involved is \( x \), making it easy to apply in various differentiation problems.

Product Rule

When you need to differentiate a function that is the product of two other functions, you apply the product rule. It is a fundamental rule in calculus stating that the derivative of the product of two functions is not simply the product of their derivatives.
The product rule is expressed as:

• \((uv)' = u'v + uv'\)

In this formula, \( u \) and \( v \) are functions of \( x \). You differentiate one function while keeping the other constant, then switch. This rule becomes crucial when handling products like \( x \cot^{-1}(x) \) in our example exercise.
Applying the product rule avoids common mistakes, ensuring you calculate derivatives of complex product functions accurately.

Arc Cotangent

The arc cotangent function, denoted as \( \cot^{-1}(x) \), is the inverse of the cotangent function, similar to how arctangent is related to tangent. The arc cotangent helps determine the angle whose cotangent is \( x \). Its range is typically from \( 0 \) to \( \pi \).
Just like the arctangent function, the derivative of arc cotangent is quite specific:

• \( \frac{d}{dx}(\cot^{-1}(x)) = \frac{-1}{1+x^2} \)

It's important to note the negative sign in the derivative formula, which is a key distinction between the derivatives of \( \arctan(x) \) and \( \arcot(x) \). This negative relationship stems from the behavior of the cotangent function over its domain.

Differentiation Rules

Differentiation rules are the guidelines or formulas used to compute derivatives of various functions. Mastery of these rules allows you to differentiate everything from simple polynomials to complex trigonometric functions efficiently.
Key differentiation rules include:

• The sum rule: \( (f + g)' = f' + g' \)
• The product rule: \( (uv)' = u'v + uv' \)
• The chain rule: Used when dealing with compositions of functions
• Derivatives of inverse trigonometric functions, like arctangent and arc cotangent

For the function \( g(x) = \tan^{-1}(x) + x\cot^{-1}(x) \), a combination of these rules comes into play, displaying the beautiful interconnectedness of differential calculus.