Question

$$ \text { } \text { find } D_{x} y . $$ $$ y=\tan ^{2} x $$

Short answer

The derivative is \( D_x y = 2 \tan x \cdot \sec^2 x \).

Step-by-step solution

Recognize the Function Form

The function given to us is \( y = \tan^2 x \). This can be rewritten using exponent rules as \( y = (\tan x)^2 \). The expression is a composite function where the outer function is \( u^2 \) with \( u = \tan x \).

Apply the Chain Rule

To differentiate \( y = (\tan x)^2 \), we will use the chain rule. The chain rule states that if \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

Differentiate the Outer Function

Let's first differentiate the outer function. If \( u = \tan x \), then the outer function is \( u^2 \). The derivative of \( u^2 \) with respect to \( u \) is \( 2u \).

Differentiate the Inner Function

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

Apply the Chain Rule Derivatives

Now apply the chain rule: \( \frac{dy}{dx} = 2u \cdot \frac{du}{dx} \). Substitute in the values: \( \frac{dy}{dx} = 2(\tan x) \cdot \sec^2 x \).

Simplify the Expression

Finally, simplify the expression: \( D_x y = 2 \tan x \cdot \sec^2 x \). This is the derivative of \( y \) with respect to \( x \).

Key concepts

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. It helps us find derivatives of functions that are composed of two or more functions. In our example, we have the function \( y = (\tan x)^2 \). This is a composite function because it contains the inner function \( \tan x \) nested within the outer function \( u^2 \).
The chain rule states: if you have a function \( y = f(g(x)) \), then the derivative is given by \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).
Applying this rule to our problem, we recognize:

• The outer function is \( u^2 \) with \( u = \tan x \).
• The inner function is \( u = \tan x \).

We begin by differentiating these parts separately, then combine them using the chain rule. This approach simplifies finding the derivative of complex, nested functions.

Trigonometric Functions

Trigonometric functions play a key role in calculus and are at the heart of this exercise. The basic trigonometric functions include sine, cosine, and tangent. Each of these functions has its own derivative, which is crucial for solving calculus problems where these functions are involved.
Tangent, or \( \tan x \), is particularly important here. It is the ratio of sine and cosine, \( \tan x = \frac{\sin x}{\cos x} \). Its derivative, \( \sec^2 x \), is derived from the chain rule applied to the trigonometric identity \( \frac{d}{dx} \left( \tan x \right) \).
Understanding the derivatives of trigonometric functions helps in accurately determining the slope of curves that are defined by trigonometric expressions. In our exercise, the derivative of \( \tan x \) is essential for applying the chain rule accurately.

Derivative of Tangent

Finding the derivative of the tangent function is a vital step in this problem. Tangent, \( \tan x \), is a primary trigonometric function. The derivative of \( \tan x \) is \( \sec^2 x \).
This derivative can be understood by considering the fact that \( \tan x = \frac{\sin x}{\cos x} \). By using the quotient rule or known derivative rules, one can derive that its slope, or instantaneous rate of change, is described by \( \sec^2 x \).
Knowing the derivative of the tangent function allows you to solve more complex differential problems where tangent is part of a larger expression. For this exercise, it allows for the correct application of the chain rule when differentiating \( y = (\tan x)^2 \).

Composite Functions

Composite functions involve one function nested inside another. The function \( y = (\tan x)^2 \) is an example where \( \tan x \) is the inner function and \( u^2 \) is the outer function. Calculus often requires finding the derivative of such composite expressions.
To differentiate a composite function, we break it down into its constituent parts. The derivative process involves:

• Identifying the inner and outer functions.
• Using the chain rule to differentiate each part separately.
• Combining these derivatives to form the overall derivative of the composite function.

In our exercise, recognizing and handling the composite nature of \( y = (\tan x)^2 \) simplifies solving it by applying the chain rule correctly, leading to \( D_x y = 2 \tan x \cdot \sec^2 x \). This process is crucial for tackling complex functions in calculus efficiently.