Problem 30 - University Calculus: Early Transcendentals

Question

Find \(d y\). $$y=2 \cot \left(\frac{1}{\sqrt{x}}\right)$$

Step-by-Step Solution

Answer

\(dy = \frac{\csc^2(1/\sqrt{x})}{x^{3/2}}\,dx\)

Step 1: Differentiate Using Chain Rule

\(y = 2\cot\left(\frac{1}{\sqrt{x}}\right) = 2\cot(x^{-1/2})\)
\(\frac{dy}{dx} = 2 \cdot (-\csc^2(x^{-1/2})) \cdot \left(-\frac{1}{2}x^{-3/2}\right) = \frac{\csc^2(x^{-1/2})}{x^{3/2}}\)

Step 2: Write dy

\(dy = \frac{\csc^2\left(\frac{1}{\sqrt{x}}\right)}{x^{3/2}}\,dx\)

Key Concepts

Chain Rule

The chain rule is an essential tool in calculus for finding the derivative of composite functions. A function is composite when you have a function within another function. In the exercise, we encounter a composite function where you have a cotangent function with an expression inside it. Here, the chain rule helps by differentiating the outer function first, leaving the inner function untouched initially. Then, we multiply that derivative by the derivative of the inner function. This adjustment accounts for the way the inner function changes the overall function.

• The outer function is identified as the cotangent, written as \( \cot(u) \), with \( u = \frac{1}{\sqrt{x}} \).
• The chain rule requires us to first differentiate the cotangent, giving \(-\csc^2(u)\).
• Next, we differentiate the inner function \( \frac{1}{\sqrt{x}} \), which involves applying our differentiation techniques.

This sequence ensures that each part of the composite function is properly addressed in finding the full derivative.

Trigonometric Functions

Trigonometric functions like sine, cosine, and cotangent are recurrent in calculus due to their cyclic nature. In this exercise, cotangent is the trigonometric function of interest. Remember, each trigonometric function has a distinct derivative:

• The derivative of \( \sin(u) \) is \( \cos(u) \).
• The derivative of \( \cos(u) \) is \( -\sin(u) \).
• The derivative of \( \tan(u) \) is \( \sec^2(u) \).
• The derivative of \( \cot(u) \) is \( -\csc^2(u) \).

These derivatives are foundational and it is vital to memorize them for quick computation in calculus problems. In the problem at hand, understanding the derivative of \( \cot(u) \) is especially important, as it allows you to take the first crucial step in differentiating the composite function.

Differentiation Techniques

Differentiation techniques are the methods used to find the derivative of a function, which represents the rate of change. Beyond typical rules like power, product, and quotient rules, the chain rule is particularly important for composites like our example function. This function requires specific techniques:

• Identify derivatives of basic functions like x to the power a, where the derivative of \( x^a \) is \( ax^{a-1} \).
• Utilize transformations: \( \frac{1}{\sqrt{x}} \) can be rewritten as \( x^{-1/2} \).
• Apply the power rule to find the derivative of the inner portion, which is \(-\frac{1}{2}x^{-3/2} \).

Efficiently using these techniques ensures accurate results and builds a solid understanding for tackling more complex calculus problems. Always remember that practice is key to mastering differentiation.