Problem 5 - Calculus Single Variable

Question

Calculate the derivative of the given expression with respect to \(x\). $$ \cos (1 / x) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{\sin(1/x)}{x^2} \).

Step 1: Identify the Outer and Inner Functions

The given expression is \( \cos(1/x) \). Here, the outer function is \( \cos(u) \) where \( u = 1/x \). The inner function is \( 1/x \). We will apply the chain rule to find the derivative.

Step 2: Find Derivative of Outer Function

Differentiate the outer function \( \cos(u) \) with respect to \( u \). The derivative is \( -\sin(u) \).

Step 3: Find Derivative of Inner Function

Differentiate the inner function \( 1/x \) with respect to \( x \). The derivative is \( -1/x^2 \).

Step 4: Apply Chain Rule

According to the chain rule, the derivative of \( \cos(1/x) \) with respect to \( x \) is the product of the derivative of the outer function at the inner function and the derivative of the inner function. This is:\[ \frac{d}{dx}(\cos(1/x)) = -\sin(1/x) \cdot \left( -\frac{1}{x^2} \right) \]

Step 5: Simplify the Expression

Multiply the expressions to get the final derivative:\[ \frac{d}{dx}(\cos(1/x)) = \frac{\sin(1/x)}{x^2} \]

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of composite functions. A composite function is one where a function is nested inside another function, like the given expression

• Outer function: \( \, \cos(u) \, \) where \( \, u = 1/x \).
• Inner function: \( \, 1/x \, \).

The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This might sound tricky at first, but when you break it down into steps, it becomes manageable.First, you'll find the derivative of each function separately. Then, multiply them together as per the chain rule: \[ \, \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x) \, \]This is all about breaking down the problem and then gradually piecing it back together.

Trigonometric Derivatives

Trigonometric functions like sine and cosine have specific derivatives that you should memorize, as they are used frequently.- The derivative of \( \sin(x) \) is \( \cos(x) \).- The derivative of \( \cos(x) \) is \( -\sin(x) \).Knowing these basic derivatives helps tackle more complex trigonometric problems, like in the given exercise.In the expression \( \cos(1/x) \), \( \cos \) is the trigonometric part. The derivative of the outer function \( \cos(u) \) with respect to \( u \) is \( -\sin(u) \). When these derivatives are applied in the context of the chain rule, they help simplify the calculation process. Understanding how basic trigonometric derivatives work lays the foundation for tackling more intricate problems effectively.

Differentiation Techniques

Differentiation is the process of finding the derivative of a function, which represents how fast the function's value is changing. Different techniques exist for differentiation, each useful in various scenarios. The exercise above involves simple differentiation, using techniques like:- **Basic Rules**: Understanding fundamental differentiation formulas, like \( \frac{d}{dx}(x^n) = nx^{n-1} \), is crucial.- **Chain Rule**: As discussed, this technique helps differentiate composite functions effectively.Moreover, in real functions involving more complexity, combining multiple differentiation rules and techniques can often lead to simplified solutions. It is vital to approach each problem by first identifying what techniques will be most effective in breaking down the function into manageable parts, just as we did with the inner and outer functions.