Problem 23 - Calculus - AP Edition

Question

$$\text { Find } d y /\left.d x\right|_{x=1}$$. $$y=\left(\frac{3 x+2}{x}\right)\left(x^{-5}+1\right)$$

Step-by-Step Solution

Answer

The derivative \(\frac{dy}{dx}\) at \(x = 1\) is -29.

Step 1: Rewrite the Function

Express the given function in a more manageable form. Starting with:\[ y = \left(\frac{3x + 2}{x}\right)\left(x^{-5}+1\right) \]Simplify the expression:\[ y = (3 + \frac{2}{x})(x^{-5} + 1) \]

Step 2: Expand the Product

Use the distributive property to expand the expression:\[ y = (3)(x^{-5} + 1) + (\frac{2}{x})(x^{-5} + 1) \]Simplify each term:\[ y = 3x^{-5} + 3 + 2x^{-5} \cdot x^{-1} + \frac{2}{x} \]Combine terms:\[ y = 3x^{-5} + \frac{2}{x^6} + 3 + \frac{2}{x} \]

Step 3: Differentiate the Function

Take the derivative with respect to \(x\) using power rule:\[ y = 3x^{-5} + \frac{2}{x^6} + 3 + \frac{2}{x} \]The derivative is:\[ \frac{dy}{dx} = -15x^{-6} - \frac{12}{x^7} - \frac{2}{x^2} \]

Step 4: Evaluate the Derivative at \(x = 1\)

Substitute \(x = 1\) into the derivative:\[ \frac{dy}{dx} = -15(1)^{-6} - \frac{12}{1^7} - \frac{2}{1^2} \]Simplify:\[ \frac{dy}{dx} = -15 - 12 - 2 \]\[ \frac{dy}{dx} = -29 \]

Key Concepts

Power Rule

The Power Rule is an essential tool in differentiation, used to find the derivative of a function with a power of a variable. It's stated quite simply: if you have a function of the form \( f(x) = x^n \), then the derivative \( f'(x) \) is \( n \cdot x^{n-1} \). This means you multiply the term by its power and decrease the power by one.

In our exercise, the Power Rule is used multiple times. For instance, for the term \( 3x^{-5} \), applying the Power Rule gives \( -15x^{-6} \), as you multiply 3 by -5 and reduce the exponent by one.

It simplifies the process of differentiating polynomial expressions, even those with negative or fractional exponents. Its simplicity makes it a favorite among calculus learners and saves time in computations.

Derivative Evaluation

Evaluating the derivative means finding the rate at which a function is changing at a certain point. In this exercise, after computing \( \frac{dy}{dx} \), the next step is to evaluate it at a specific value of \( x \).

Once the derivative of our function is found, \( \frac{dy}{dx} = -15x^{-6} - \frac{12}{x^7} - \frac{2}{x^2} \), we substitute \( x = 1 \) to find how the function behaves at that point. This step involves simple substitution followed by arithmetic calculations.

For \( x = 1 \), this leads to \( \frac{dy}{dx} = -15(1)^{-6} - \frac{12}{1^7} - \frac{2}{1^2} = -15 - 12 - 2 = -29 \).

Understanding this allows students to interpret the derivative meaningfully, seeing how the changes in \( x \) directly affect the function.

Algebraic Manipulation

Before we can apply calculus, sometimes algebraic manipulation is necessary. This involves simplifying expressions or changing their form to make differentiation or integration easier.

In this problem, the expression for \( y \) is rewritten by separating components and expanding them using distributive property, transforming \( (\frac{3x + 2}{x})(x^{-5} + 1) \) into a simplified version \( 3x^{-5} + \frac{2}{x^6} + 3 + \frac{2}{x} \).

An important ability in algebraic manipulation is combining like terms, such as \( 3x^{-5} \) with other terms having the same form. This step is crucial to set the stage for a smoother differentiation process.

Learning such skills not only helps in calculus but also establishes a solid foundation for problem-solving in various branches of mathematics.