Problem 25 - University Calculus: Early Transcendentals

Question

Find \(d y\). $$y=\sin (5 \sqrt{x})$$

Step-by-Step Solution

Answer

The differential is \( dy = \frac{5 \cos(5 \sqrt{x})}{2 \sqrt{x}} dx \).

Step 1: Identify the Function and Variables

The given function is \( y = \sin(5 \sqrt{x}) \). Here, \( y \) is a function of \( x \). We need to differentiate \( y \) with respect to \( x \) to find \( dy \).

Step 2: Use the Chain Rule

The function \( y = \sin(5 \sqrt{x}) \) is a composite function. To differentiate it, we use the chain rule. The outer function is \( \sin(u) \), where \( u = 5 \sqrt{x} \). The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \).

Step 3: Differentiate the Outer Function

Differentiate \( \sin(5 \sqrt{x}) \) with respect to \( 5 \sqrt{x} \) using \( \frac{d}{du} \sin(u) = \cos(u) \). This gives us \( \cos(5 \sqrt{x}) \).

Step 4: Differentiate the Inner Function

Next, differentiate the inner function \( 5 \sqrt{x} = 5x^{1/2} \) with respect to \( x \). Using the power rule, \( \frac{d}{dx} x^{1/2} = \frac{1}{2}x^{-1/2} \), we get \( \frac{d}{dx} 5\sqrt{x} = 5 \times \frac{1}{2}x^{-1/2} = \frac{5}{2\sqrt{x}} \).

Step 5: Apply the Chain Rule

Combine the results from Step 3 and Step 4 to apply the chain rule: \[ \frac{dy}{dx} = \cos(5 \sqrt{x}) \cdot \frac{5}{2 \sqrt{x}} \].

Step 6: Simplify the Expression

Simplify the expression to get \( \frac{dy}{dx} = \frac{5 \cos(5 \sqrt{x})}{2 \sqrt{x}} \). This is the derivative of \( y \) with respect to \( x \).

Step 7: Express in Terms of Differential

The differential \( dy \) can be expressed as \( dy = \frac{5 \cos(5 \sqrt{x})}{2 \sqrt{x}} dx \).

Key Concepts

Chain Rule

The Chain Rule is a fundamental technique in calculus for differentiating compositions of functions. When you have a function nested within another, as in the case of the composite function problem, the Chain Rule helps you differentiate efficiently.
For example, when faced with differentiating something like \(y = \sin(5\sqrt{x})\), the Chain Rule tells us to first look at the outermost function, which is \(\sin(u)\), where \(u = 5\sqrt{x}\).

• The derivative of \(\sin(u)\) with respect to \(u\) is \(\cos(u)\).
• Then, you multiply this by the derivative of the inner function, \(5\sqrt{x}\), with respect to \(x\).

This straightforward yet powerful method allows breaking down complex expressions into easier, manageable pieces that can be differentiated step by step.

Composite Function

A Composite Function is essentially a function within a function. It is represented in calculus as \(f(g(x))\).
In our exercise, the composite function is \(\sin(5\sqrt{x})\). Here, the sine function is applied to \(5\sqrt{x}\), which makes \(\sqrt{x}\) the inner component.

• By breaking down the problem into parts: the outer function \(\sin(u)\) and the inner function \(u = 5\sqrt{x}\), the differentiation process becomes straightforward.
• Handling each part individually helps in understanding the sequence of operations, simplifying calculus problem solving.

Understanding composite functions is crucial for using the Chain Rule effectively since it allows clear identification of what constitutes the inner and outer functions.

Power Rule

The Power Rule is a basic principle in calculus that is invaluable when differentiating functions of the form \(x^n\).
It states that if you have a function \(f(x) = x^n\), its derivative, \(f'(x)\), will be \(nx^{n-1}\).
This rule is particularly useful in our example for differentiating \(\sqrt{x}\), which can be rewritten as \(x^{1/2}\).

• Applying the power rule, we find that \(\frac{d}{dx} x^{1/2} = \frac{1}{2} x^{-1/2}\).
• This allows swift calculation for components of composite functions when applying the chain rule.

Knowing the Power Rule makes solving calculus problems quicker and more intuitive, as you can directly handle any powers of \(x\) included in your function.

Calculus Problem Solving

Solving Calculus Problems involves multiple strategies and techniques. Identifying composite functions, correctly applying the Chain Rule, and using the Power Rule are all steps in a systematic approach.
In our original exercise:

• Identify the composite nature of \(\sin(5\sqrt{x})\).
• Break it into parts to efficiently apply the rules of differentiation.

This clear methodology is vital:

• Recognize the structure (composite function) of the problem.
• Select appropriate differentiation strategies (Chain Rule, Power Rule).
• Conduct calculations step by step while checking for simplifications.

Mastering these techniques not only boosts proficiency in solving such problems but also enhances overall math intuition and competency.