Problem 63 - Thomas Calculus in SI Units

Question

Find the derivative of \(y\) with respect to the given independent variable. $$y=7^{\sec \theta} \ln 7$$

Step-by-Step Solution

Answer

\( \frac{dy}{d\theta} = 7^{\sec \theta} (\ln 7)^2 \sec \theta \tan \theta \)

Step 1: Identify the Function Form

The given expression is \( y = 7^{\sec \theta} \ln 7 \). The expression is a function of \( \theta \). We need to find the derivative of \( y \) with respect to \( \theta \), noticing it is an exponential function where the exponent is a function of \( \theta \).

Step 2: Differentiate Using the Exponential Rule

Recall that the derivative of \( a^u \) with respect to \( x \) is \( a^u \ln a \cdot \frac{du}{dx} \), where \( u \) is a function of \( x \). For \( y = 7^{\sec \theta} \ln 7 \), let \( u = \sec \theta \). Then, \( y = a^u \) where \( a = 7 \).

Step 3: Apply the Chain Rule

The chain rule shows that \( \frac{dy}{d\theta} = 7^{\sec \theta} \ln 7 \cdot \ln 7 \cdot \frac{d}{d\theta}(\sec \theta) \). This factorizes into two parts: differentiating the exponent with respect to \( \theta \).

Step 4: Differentiate \( \sec \theta \)

Find the derivative of \( \sec \theta \) with respect to \( \theta \). The derivative is \( \sec \theta \tan \theta \).

Step 5: Combine Results

Substitute back the derivative of \( \sec \theta \) and compute \( \frac{dy}{d\theta} = 7^{\sec \theta} (\ln 7)^2 \cdot \sec \theta \tan \theta \).

Step 6: Simplify the Expression

Combine all terms to get the simplified expression: \( \frac{dy}{d\theta} = 7^{\sec \theta} (\ln 7)^2 \sec \theta \tan \theta \).

Key Concepts

Exponential Functions

Exponential functions are an important class of mathematical functions where a constant base is raised to a variable exponent. They have various applications in fields such as science, economics, and engineering. For the function \( y = a^x \), the base \( a \) is a constant, while \( x \) is the variable exponent. The exponential function grows rapidly as the exponent increases, making it useful for modeling growth processes such as population growth or radioactive decay.

When dealing with exponential functions like \( y = 7^{\sec \theta} \ln 7 \), understanding its derivative is crucial, especially as these functions transform into useful shapes by varying the exponent. In calculus, the derivative of an exponential function is complex, requiring the use of specific rules and the chain rule, particularly when the exponent itself is another function, such as \( \sec \theta \).

• Base: In the function \( 7^{\sec \theta} \), 7 is the constant base.
• Exponent: \( \sec \theta \) is the exponent, which varies based on \( \theta \).

To differentiate, it's important to remember that the rate of change of an exponential function involves multiplying by the natural logarithm of the base, \( \ln a \), and the derivative of the exponent with respect to the variable of differentiation.

Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of composite functions. When a function is composed of two or more simpler functions, this rule is invaluable. It helps compute the derivative of a function like \( y = 7^{\sec \theta} \ln 7 \), where \( 7^{u} \) is an exponential function with exponent \( u = \sec \theta \).

Utilizing the chain rule, we can break down complex derivatives into manageable steps. Let's see how the chain rule plays out here:

• First, differentiate the outer function, holding the inner function constant. So, for \( a^u \), the derivative is \( a^u \cdot \ln a \).
• Next, differentiate the inner function. In this case, \( \sec \theta \) becomes \( \sec \theta \tan \theta \).

The chain rule allows us to combine these steps to find the overall derivative. So, the derivative of \( y = 7^{\sec \theta} \ln 7 \) with respect to \( \theta \) is found by multiplying the derivative of the outer function with that of the inner function.
This results in: \( \frac{dy}{d\theta} = 7^{\sec \theta} (\ln 7)^2 \sec \theta \tan \theta \). The chain rule simplifies handling complicated calculations.

Trigonometric Derivatives

Trigonometric derivatives involve differentiating functions that include trigonometric functions like sine, cosine, and secant. These derivatives are essential for solving many calculus problems involving periodic and oscillatory functions.

Secant function \( \sec \theta = \frac{1}{\cos \theta} \) is particularly interesting due to its derivative. To find the derivative of \( \sec \theta \), basic trigonometric identities and derivatives are used:

• Derivative of \( \sec \theta \) is \( \sec \theta \tan \theta \).
• This result emerges from considering the derivative of \( \cos \theta \) and using the quotient rule.

Recognizing the derivatives of trigonometric functions is critical to applying the chain rule effectively. When you encounter a function such as \( y = 7^{\sec \theta} \ln 7 \), knowing that \( \sec \theta \) differentiates into \( \sec \theta \tan \theta \) is vital to simplify and solve the problem.
Trigonometric derivatives extend the capabilities of mathematical analysis, aiding in the understanding of functions that map angles to ratios, essential in fields like physics and engineering.