Problem 236 - Calculus Volume 1

Question

Find \(\frac{d y}{d x}\) for each function. $$ y=\sqrt{6+\sec \pi x^{2}} $$

Step-by-Step Solution

Answer

\[ \frac{d y}{d x} = \pi x \sec(\pi x^2) \tan(\pi x^2) (6+\sec(\pi x^2))^{-1/2} \]

Step 1: Identify the function structure

The given function is \( y = \sqrt{6 + \sec(\pi x^2)} \). This can be re-written as \( y = (6 + \sec(\pi x^2))^{1/2} \). It shows the combination of a constant, a trigonometric function, and a power function.

Step 2: Apply the chain rule

To find the derivative, we use the chain rule. We will first treat \( (6 + \sec(\pi x^2))^{1/2} \) as a whole and differentiate it, then multiply by the derivative of the inside part \( 6 + \sec(\pi x^2) \).

Step 3: Differentiate the outer function

Applying the power rule as part of the chain rule to differentiate \( (6 + \sec(\pi x^2))^{1/2} \), we get: \[ \frac{1}{2}(6 + \sec(\pi x^2))^{-1/2} \times \frac{d}{dx}(6 + \sec(\pi x^2)) \]

Step 4: Differentiate the inner function

Now, differentiate \( 6 + \sec(\pi x^2) \). The derivative of 6 is zero. For \( \sec(\pi x^2) \), use: \[ \frac{d}{dx} \sec(u) = \sec(u) \tan(u) \frac{du}{dx} \] where \( u = \pi x^2 \).

Step 5: Differentiate \( \pi x^2 \)

Calculate the derivative of \( u = \pi x^2 \). \[ \frac{d}{dx}(\pi x^2) = 2\pi x \]

Step 6: Combine derivatives

Substituting back in: \( \frac{d}{dx}(\sec(\pi x^2)) = \sec(\pi x^2) \tan(\pi x^2) \times 2\pi x \)Now substitute into the chain rule expression: \[ \frac{d y}{d x} = \frac{1}{2}(6 + \sec(\pi x^2))^{-1/2} \cdot \sec(\pi x^2) \tan(\pi x^2) \cdot 2\pi x \]

Step 7: Simplify the expression

Combine all parts of the expression: \[ \frac{d y}{d x} = \pi x \sec(\pi x^2) \tan(\pi x^2) \cdot (6 + \sec(\pi x^2))^{-1/2} \] Thus, the derivative of the function is fully simplified.

Key Concepts

Chain Rule in Calculus

The chain rule is a fundamental technique in calculus used for finding the derivative of composite functions. When you encounter a function that is made up of another function within it, like in our case: \( y = (6 + \sec(\pi x^2))^{1/2} \), the chain rule is your go-to tool.
The essence of the chain rule lies in differentiation of a composed function. First, you differentiate the outer function while treating the inner function as a single entity. Then, you multiply by the derivative of the inner function that you just held constant during the first step.
This can be visualized as: If \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \). Applying it to our function means differentiating the power of \( 1/2 \) separately and then multiplying by the derivative of \( 6 + \sec(\pi x^2) \). This systematic approach allows us to handle complex functions piece by piece, breaking them down into manageable parts.
Mastering the chain rule can greatly simplify derivatives involving layers of functions.

Trigonometric Functions and Their Derivatives

Trigonometric functions, such as sine, cosine, or secant, are periodic functions frequently appearing in calculus. In our exercise, \( \sec(\pi x^2) \) is the trigonometric part of the function being differentiated. To differentiate \( \sec(u) \), where \( u \) is any function of \( x \), use \( \frac{d}{dx} \sec(u) = \sec(u) \tan(u) \frac{du}{dx} \).
This formula arises from the derivative rules of trigonometric functions, specifically leveraging the secant and tangent functions' interplay in calculus.

• \( \sec(x) \) leads to \( \sec(x)\tan(x) \) after differentiation.
• The "\( \tan(u) \)" part comes from recognizing the derivative form of \( \sec(u) \).

In our scenario, u is \( \pi x^2 \), making it necessary to also apply the derivative to \( \pi x^2 \) itself. Understanding these rules ensures you correctly handle the transformation of angles and their derivatives, especially when embedded in functions.

Understanding the Power Rule

The power rule is one of the simplest yet powerful rules in calculus for differentiation. When you have a function of the form \( x^n \), its derivative is \( nx^{n-1} \). This rule easily extends to cases where the exponent \( n \) is not just a simple integer.
In the current context, we're dealing with an outer function of \( (6 + \sec(\pi x^2))^{1/2} \). When applying the derivative, it turns into \( \frac{1}{2}(6 + \sec(\pi x^2))^{-1/2} \), showcasing the power rule's adaptation with fractional exponents.
Here's a quick step-by-step:

• Identify the exponent, which is \( 1/2 \).
• Bring down \( 1/2 \) as a coefficient and subtract 1 from the original exponent.

The subtraction leads to a new power of \(-1/2\). This change of sign indicates a flip between square root and reciprocal function behavior, affecting subsequent simplifications. The power rule's flexibility allows for straightforward differentiation, even in composite function scenarios.