Question

Compute the derivative of the given function. $$g(x)=\tan ^{-1}(2 x)$$

Short answer

The derivative is \( g'(x) = \frac{2}{1+4x^2} \).

Step-by-step solution

Identify the Function Type

The function given is \( g(x) = \tan^{-1}(2x) \). This is an inverse trigonometric function, specifically the inverse tangent or arctan function.

Apply the Chain Rule

To find the derivative of \( g(x) = \tan^{-1}(2x) \), we'll apply the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is \( f'(g(x)) \cdot g'(x) \).

Differentiate the Outer Function

The derivative of \( \tan^{-1}(u) \) with respect to \( u \) is \( \frac{1}{1+u^2} \). For the function \( g(x) = \tan^{-1}(2x) \), let \( u = 2x \). Then, the derivative of the outer function with respect to \( u \) is \( \frac{1}{1+(2x)^2} \).

Differentiate the Inner Function

Now, find the derivative of the inner function, \( u = 2x \). The derivative of \( 2x \) with respect to \( x \) is 2.

Combine Derivatives Using the Chain Rule

Combine the derivatives from Step 3 and Step 4 using the chain rule. Multiply the derivative of the outer function \( \frac{1}{1+(2x)^2} \) by the derivative of the inner function, which is 2. This gives: \[ g'(x) = \frac{1}{1+(2x)^2} \cdot 2 = \frac{2}{1+4x^2} \]

Simplify the Expression

The expression \( \frac{2}{1+4x^2} \) is already simplified. This is the derivative of the function \( g(x) = \tan^{-1}(2x) \).

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus used for finding the derivative of composite functions. When you have a function composed of other functions, such as \( f(g(x)) \), the chain rule provides a method to find its derivative. For a function of the form \( f(g(x)) \), the chain rule states that

• differentiate the outer function \( f \) with respect to \( g \),
• then multiply by the derivative of the inner function \( g \) with respect to \( x \).

This can be represented as \( \frac{df}{dx} = f'(g(x)) \cdot g'(x) \).
To apply the chain rule effectively, identifying the inner and outer functions is crucial. In this exercise, \( g(x) = \tan^{-1}(2x) \), the outer function is the inverse tangent, and the inner function is \( 2x \). Using the chain rule helps in breaking down complex differentiation into manageable steps, ultimately leading to accurate calculation of a derivative.

Derivative of Arctan

The derivative of the inverse tangent function, often denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \), is a critical concept in calculus. The derivative is given by:\[\frac{d}{dx}[\tan^{-1}(x)] = \frac{1}{1+x^2}\]This derivative is fundamental when dealing with functions involving inverse trigonometric components.
When you apply this to a composite function, such as \( \tan^{-1}(2x) \), the derivative is first calculated concerning the variable inside the inverse function. Thus, with \( u = 2x \), the derivative of \( \tan^{-1}(u) \) with respect to \( u \) is \( \frac{1}{1+u^2} \).
This insight into the derivative of the arctan function simplifies finding the rate of change for functions involving arctangent, making it a useful tool across various calculus problems.

Composite Functions

Composite functions occur when one function is applied to the result of another function. Writing a function as a composition means expressing it in the form \( h(x) = f(g(x)) \), where \( g(x) \) is applied first, and then \( f \) is applied to the result of \( g(x) \).
In our example, \( g(x) = \tan^{-1}(2x) \), the function can be seen as a composition where the outer function is \( \tan^{-1}(x) \) and the inner function is \( 2x \).
Composite functions allow for complex function relationships to be modeled effectively and require careful use of techniques like the chain rule for differentiation. When faced with differentiating composite functions:

• first find the derivative of the outer function,
• then the inner function,
• finally, apply the chain rule to piece them together.

Breaking down complex functions into simpler parts can make analysis and problem-solving more manageable.

Differentiation Techniques

Differentiation techniques are various methods used to find the derivative of a function. Different problems require selecting the right technique to simplify the differentiation process.
Some commonly used techniques include the product rule, quotient rule, and chain rule. The choice of technique largely depends on the form of the function. In cases like \( g(x) = \tan^{-1}(2x) \), the chain rule becomes essential due to the composition of functions. In contrast:

• The product rule is used when differentiating products of functions, \( (uv)' = u'v + uv' \).
• The quotient rule applies to the division of functions, \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \).

Mastering these techniques allows you to approach a wide variety of functions with confidence, ensuring you can effectively find derivatives in any situation.