Problem 34 - Calculus Single Variable

Question

Use the Quotient Rule to compute the derivative of the given expression with respect to \(x .\) $$ \cos (x) /(x-5) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{-x\sin(x) + 5\sin(x) - \cos(x)}{(x-5)^2} \).

Step 1: Identify Functions

The expression \( \frac{\cos(x)}{x-5} \) can be seen as a quotient \( \frac{u}{v} \), where \( u(x) = \cos(x) \) and \( v(x) = x - 5 \).

Step 2: Recall the Quotient Rule

The Quotient Rule states that the derivative of a quotient \( \frac{u}{v} \) is given by \( \frac{u'v - uv'}{v^2} \).

Step 3: Find the Derivatives of u and v

Calculate the derivative of \( u(x) = \cos(x) \), which is \( u'(x) = -\sin(x) \). For \( v(x) = x - 5 \), the derivative \( v'(x) = 1 \).

Step 4: Apply the Quotient Rule

Substitute the derivatives and the original functions into the Quotient Rule: \[ \frac{d}{dx}\left(\frac{\cos(x)}{x-5}\right) = \frac{(-\sin(x))(x-5) - (\cos(x))(1)}{(x-5)^2} \].

Step 5: Simplify the Expression

Simplify the expression to obtain the derivative: \[ \frac{-\sin(x)(x-5) - \cos(x)}{(x-5)^2} = \frac{-x\sin(x) + 5\sin(x) - \cos(x)}{(x-5)^2} \].

Key Concepts

Quotient Rule

In calculus, the Quotient Rule is a vital technique for finding the derivative of a function that is the division of two other functions. The rule provides an effective way to handle cases where you have a function expressed as a ratio, \ \( \frac{u}{v} \ \).The formula for the Quotient Rule is:- \ \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \ \)Where:- \ \( u \ \) and \ \( v \ \) are functions of \ \( x \ \)- \ \( u' \ \) is the derivative of \ \( u \ \)- \ \( v' \ \) is the derivative of \ \( v \ \)This rule helps manage the complexity that can arise in division problems, especially when the numerator and denominator are both non-linear functions. It ensures that all aspects of both functions are appropriately considered in the differentiation process. For example, if you have a function like \ \( \frac{\cos(x)}{x-5} \ \), identifying your \ \( u \ \) and \ \( v \ \) and applying the rule is straight-forward, as demonstrated in the exercise.

Derivative

The concept of a derivative is fundamental in calculus. The derivative measures how a function changes as its input changes, often referred to as the "rate of change" or "slope" of the function. In mathematical notation, the derivative of a function \ \( f(x) \ \) with respect to \ \( x \ \) is often written as \ \( f'(x) \ \) or \ \( \frac{df}{dx} \ \).To calculate the derivative, you need to identify how each part of the function changes with small changes in \ \( x \ \):- A constant term has a derivative of 0 since it doesn't change.- A function like \ \( \cos(x) \ \) varies with \ \( x \ \), leading to a derivative of \ \( -\sin(x) \ \).The derivative is what allows us to apply rules like the Quotient Rule to more complex functions beyond simple polynomials. \ \[ \cos(x) \ \] changing to \ \[ -\sin(x) \ \] is a common result you often deal with in calculus when working with trigonometric functions.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are essential in calculus, especially when differentiated or integrated. They represent fundamental patterns of oscillation and rotation, making them applicable in various scientific fields.For example, the functions important to our problem are:- \ \( \cos(x) \ \) which describes the horizontal component of a rotating unit circle.The derivatives of these trigonometric functions are crucial for solving calculus problems:- The derivative of \ \( \cos(x) \ \) is \ \( -\sin(x) \ \). This tells us how \ \( \cos(x) \ \) changes at any given point \ \( x \ \).- Similarly, other trigonometric identities often help to simplify expressions after differentiation.In problems like the one given, understanding the roles and transformations of trigonometric functions is key to accurately applying techniques such as the Quotient Rule.

Simplification

After applying calculus rules and differentiation techniques, simplification is an important step. This process involves making complex expressions more understandable and often more manageable for further analysis or computation.In the context of derivatives, especially when using the Quotient Rule, you often arrive at a result that can be quite intricate. For instance:- We began with \ \( \frac{-\sin(x)(x-5) - \cos(x)}{(x-5)^2} \ \)- Through some algebraic simplifications, we can reorganize terms: \ - Start by expanding products: \ \(-x\sin(x) + 5\sin(x) - \cos(x)\ \).Simplifying helps you get a cleaner and often more insightful form of the result, making it easier to comprehend and use in further applications. This step, sometimes overlooked, can highlight symmetries or patterns not initially obvious in the original complex formulation.