Question

Find \(d y / d x\) implicitly and find the largest interval of the form \(-a< y< a\) or \(0< y< a\) such that \(y\) is a differentiable function of \(x .\) Write \(d y / d x\) as a function of \(x .\) \(\tan y=x\)

Short answer

The derivative \( \frac{dy}{dx} = \frac{1}{1 + x^2} \) and the largest interval for \( y \) is \( -\frac{\pi}{2} < y < \frac{\pi}{2} \).

Step-by-step solution

Implicit Differentiation

Implicitly differentiate both sides of the function with respect to \( x \). Applying the chain rule on the left side, we know that the derivative of \( \tan y \) is \( \sec^2 y \) times \( \frac{dy}{dx} \), on other hand, the derivative of \( x \) with respect to \( x \) is \( 1 \). Hence we get \( \sec^2 y \cdot \frac{dy}{dx} = 1 \)

Solve for \(\frac{dy}{dx}\)

To isolate \( \frac{dy}{dx} \) rearrange the equation: \[ \frac{dy}{dx} = \frac{1}{\sec^2 y} \] and knowing \( \sec^2 y = 1 + \tan^2 y \) we substitute \( x \) for \( \tan y \) on the right to get: \[ \frac{dy}{dx} = \frac{1}{1 + x^2} \]

Determine the Largest Interval

Since \( \tan y = x \) and \( \tan y \) is differentiable for all real numbers except at points where \( y = (2n+1)\frac{\pi}{2} \), where \( n \) is an integer. This occurs when \( y = \frac{\pi}{2} \) and \( y = -\frac{\pi}{2} \), so the largest interval is \( -\frac{\pi}{2} < y < \frac{\pi}{2} \)

Key concepts

Chain Rule

Understanding the chain rule is crucial for implicit differentiation. When dealing with functions that are not in the standard form of function equals variable, implicit differentiation is required. The chain rule helps integrate the relationship between two functions where one is a function of another. For example, in our exercise where we have \( \tan y = x \), both \( y \) and \( x \) are variables. To differentiate \( \tan y \) with respect to \( x \), we apply the chain rule. This rule tells us that the derivative of \( \tan y \) is \( \sec^2 y \) multiplied by \( \frac{dy}{dx} \). Hence, the application of the chain rule allows us to transform an implicit function into a more solvable derivative equation. This technique is foundational for solving the problem of finding \( \frac{dy}{dx} \) in implicit situations.

Secant Function

The secant function is integral to understanding the derivative of the tangent function, \( \tan y \). This relationship is crucial in the context of differentiation.The secant function, denoted as \( \sec y \), is defined as \( \sec y = \frac{1}{\cos y} \). When differentiating \( \tan y \), it results in \( \sec^2 y \), the square of the secant function. This is important as \( \sec^2 y \) arises directly when using the chain rule for implicit differentiation in functions involving tangent, as seen in the exercise. This outcome allows us to solve for \( \frac{dy}{dx} \) and proceed to find the derivative efficiently.

Tangent Function

The tangent function, \( \tan y \), is one of the primary trigonometric functions, often leading to unique differentiation challenges due to its properties.In the exercise, we start with \( \tan y = x \). The derivative of \( \tan y \) is \( \sec^2 y \), illustrating how differentiation applies to trigonometric functions. It’s essential to note that \( \tan y \) has discontinuities. For differentiable purposes, it means angles like \( \frac{\pi}{2} \), where tangent goes to infinity, must be excluded from the validity interval. These properties explain why we must determine allowable intervals for differentiation, which leads us to understanding the interval of differentiation.

Interval of Differentiation

When determining the interval of differentiation, we must consider where the function involved is continuous and differentiable. For the tangent function, \( \tan y = x \), it is differentiable everywhere except where the secant becomes undefined.Since \( \tan y \) becomes undefined at angles like \( y = \frac{(2n+1)\pi}{2} \), where \( n \) is an integer, these points introduce discontinuities. Therefore, the largest differentiable interval for \( y \) is any subset where \( -\frac{\pi}{2} < y < \frac{\pi}{2} \). Staying within these bounds ensures \( \tan y \) does not hit asymptotes reflected in a lack of continuity. Maintaining differentiability across \( x \'s \) real number line with this guidance allows correct implicit differentiation.