Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\). $$ y=\left(\tan ^{-1} x\right)^{3} $$

Step-by-Step Solution

Answer

\(D_x y = \frac{3(\tan^{-1} x)^2}{1+x^2}\)

Step 1: Set Up the Derivative

We are given the function \(y = (\tan^{-1} x)^3\). To find \(D_x y\), we need to use the chain rule because \(y\) is composed of an outer and an inner function.

Step 2: Differentiate the Outer Function

The outer function is \(u^3\) where \(u = \tan^{-1} x\). The derivative of \(u^3\) with respect to \(u\) is \(3u^2\).

Step 3: Differentiate the Inner Function

The inner function \(u = \tan^{-1} x\) has the derivative \(\frac{d}{dx} (\tan^{-1} x) = \frac{1}{1+x^2}\).

Step 4: Apply the Chain Rule

According to the chain rule, the derivative \(D_x y\) is found by multiplying the derivative of the outer function by the derivative of the inner function. This gives us: \[ \frac{dy}{dx} = 3(\tan^{-1} x)^2 \cdot \frac{1}{1+x^2} \]

Step 5: Simplify the Expression

Now, we simplify the expression we obtained:\[ \frac{dy}{dx} = \frac{3(\tan^{-1} x)^2}{1+x^2} \]This is the simplified form of the derivative.

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate compositions of functions. Think of it as a process to tackle functions within functions. This is especially useful when a function is nested, like layers of an onion. In the original exercise, our function is composed of an outer function, raised to the power of three, and an inner function, which is the arctan function.Here's how the chain rule works:

• First, identify the outer function and find its derivative. In our example, if you have something like \(u^3\), the derivative with respect to \(u\) is \(3u^2\).
• Next, differentiate the inner function separately. For \(u = \tan^{-1} x\), its derivative is \(\frac{1}{1+x^2}\).
• Finally, to find the derivative of the whole composition, multiply the two derivatives together. This ensures you're accounting for both functions correctly.

Using these steps, the chain rule finds derivatives efficiently, even for complex nested functions.

Inverse Trigonometric Functions

Inverse trigonometric functions, like \(\tan^{-1} x\), are essential when reversing trigonometric ratios. They answer questions such as: "What angle yields this value for tangent?" These functions are celebrated for returning angles, rather than ratios, which is a crucial distinction.When differentiating inverse trigonometric functions, each function has a specific derivative:

• The derivative of \(\tan^{-1} x\) is \(\frac{1}{1+x^2}\).
• Other functions, like \(\sin^{-1} x\) and \(\cos^{-1} x\), have their unique derivatives.

Understanding these derivatives allows us to solve problems involving inverse functions, as demonstrated in the exercise. By utilizing these derivatives, one can address more complex calculus challenges effectively.

Derivative Simplification

Simplifying derivatives is often the final step in the differentiation process. It's about making the expression more readable and manageable. After applying the chain rule, derivatives can sometimes seem complex and cluttered.Here's why simplification is important:

• Reduces the complexity of the expression.
• Makes the result easier to interpret and apply.
• Prepares the derivative for further mathematical processes or evaluations.

In the original exercise, after applying the chain rule, the derivative was initially expressed as \(3(\tan^{-1} x)^2 \cdot \frac{1}{1+x^2}\). Simplifying it leads to \(\frac{3(\tan^{-1} x)^2}{1+x^2}\), which is cleaner and often easier to work with in subsequent calculations. Always aim for simplicity in mathematics, as it often leads to greater clarity and efficiency.