Problem 63 - University Calculus: Early Transcendentals

Question

In Exercises \(51-70,\) find \(d y / d t\). $$y=\left(1+\tan ^{4}\left(\frac{t}{12}\right)\right)^{3}$$

Step-by-Step Solution

Answer

The derivative is \(\frac{1}{4}(1+\tan^4\left(\frac{t}{12}\right))^2 \cdot \tan^3\left(\frac{t}{12}\right) \cdot \sec^2\left(\frac{t}{12}\right)\).

Step 1: Identify the Chain Rule

To differentiate the function \(y = \left(1+\tan^4\left(\frac{t}{12}\right)\right)^3\), we'll need to apply the chain rule multiple times. This function is a composition of the outer function \(u^3\) and the inner function \(u = 1 + \tan^4\left(\frac{t}{12}\right)\).

Step 2: Differentiate the Outer Function

The outer function is \(u^3\). Using the power rule, the derivative is \(3u^2\). Thus, we have \(\frac{du^3}{du} = 3u^2\), where \(u = 1 + \tan^4\left(\frac{t}{12}\right)\).

Step 3: Differentiate the Inner Function

Now, differentiate the inner function \(u = 1 + \tan^4\left(\frac{t}{12}\right)\). First, derive \(\tan^4\left(\frac{t}{12}\right)\) which is \(4\tan^3\left(\frac{t}{12}\right) \cdot \sec^2\left(\frac{t}{12}\right) \cdot \frac{d}{dt}\left(\frac{t}{12}\right)\).

Step 4: Derive \(\tan\left(\frac{t}{12}\right)\)

The derivative \(\frac{d}{dt}\left(\frac{t}{12}\right)\) is \(\frac{1}{12}\). So the derivative of \(u = 1 + \tan^4\left(\frac{t}{12}\right)\) with respect to \(t\) is \(4\tan^3\left(\frac{t}{12}\right) \cdot \sec^2\left(\frac{t}{12}\right) \cdot \frac{1}{12}\).

Step 5: Apply the Chain Rule

Using the chain rule, \(\frac{dy}{dt} = \frac{dy}{du} \cdot \frac{du}{dt}\). We found \(\frac{dy}{du} = 3u^2\) and \(\frac{du}{dt} = 4\tan^3\left(\frac{t}{12}\right) \cdot \sec^2\left(\frac{t}{12}\right) \cdot \frac{1}{12}\). Now multiply these derivatives: \(3(1+\tan^4\left(\frac{t}{12}\right))^2 \cdot 4\tan^3\left(\frac{t}{12}\right) \cdot \sec^2\left(\frac{t}{12}\right) \cdot \frac{1}{12}\).

Step 6: Simplify

Simplify the expression: \(\frac{dy}{dt} = \frac{1}{4}(1+\tan^4\left(\frac{t}{12}\right))^2 \cdot \tan^3\left(\frac{t}{12}\right) \cdot \sec^2\left(\frac{t}{12}\right)\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, focusing on how a function changes as its input changes. It provides the rate at which a quantity changes, which can be particularly useful in real-world scenarios such as motion or growth problems. Let's look at how it applies to the function given in the exercise.We have a function, \(y = (1+\tan^4(\frac{t}{12}))^3\), which is a composite function. To differentiate this, we employ the chain rule. This rule states that to find the derivative of a composite function, you differentiate the outer function in terms of the inner function, and then multiply by the derivative of the inner function.In this example, the outer function is \(u^3\), where \(u = 1 + \tan^4(\frac{t}{12})\). Differentiating \(u^3\) with respect to \(u\) using the power rule, we obtain \(3u^2\). This indicates that as \(u\) changes, \(y\) changes by a factor of \(3u^2\). Following this, we move to the crucial task of differentiating the inner function.

Trigonometric Functions

Trigonometric functions like tangent (tan) and secant (sec) are pivotal in many calculus problems, especially those involving periodic phenomena like waves and oscillations.In the exercise, we're dealing with \(\tan(\frac{t}{12})\), which forms part of the composite function. When differentiating a function such as \(\tan^4(\frac{t}{12})\), we apply both the chain rule and power rule.First, differentiate \(\tan(\frac{t}{12})\) to obtain its derivative: \(\sec^2(\frac{t}{12})\). Then, using the power rule, the derivative of \(\tan^4(x)\) is \(4\tan^3(x)\). So, \(\tan^4(\frac{t}{12})\) differentiates to \(4\tan^3(\frac{t}{12}) \cdot \sec^2(\frac{t}{12})\), with an additional term of \(\frac{1}{12}\) from the derivative of \(\frac{t}{12}\).Understanding these trigonometric functions and their derivatives is essential to accurately compute the rate of change in various scenarios, ensuring comprehension in complex calculus problems.

Calculus Problem-Solving

Solving calculus problems often involves a structured approach, breaking down complex functions into simpler parts, and then systematically solving them.With the given exercise, our goal is to find \(\frac{dy}{dt}\). The steps involve:

• Applying the chain rule to differentiate the composite function, breaking it down into outer and inner functions.
• Identifying the derivatives of the trigonometric component \(\tan^4(\frac{t}{12})\) using rules for trigonometric functions.
• Combining these derivatives methodically. After computing \(\frac{dy}{du}\) and \(\frac{du}{dt}\), multiply them to get \(\frac{dy}{dt}\).

Each step requires careful application of calculus principles. Once the derivatives are found, combining and simplifying the expressions gives the final derivative form. The solution becomes a blend of algebraic manipulation and calculus techniques, each building on foundational knowledge to solve real-world problems effectively.