Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Find \(d y / d u, d u / d x\), and \(d y / d x\). $$ y=u^{-1}, u=x^{3}+2 x^{2} $$

Step-by-Step Solution

Answer

The derivatives are \(d y / d u = - u^{-2}\), \(d u / d x = 3 x^{2} + 4x\), and \(d y / d x = - (x^{3}+2 x^{2})^{-2} * (3 x^{2} + 4x)\).

Step 1: Find \(d y / d u\)

We're given \(y=u^{-1}\), or in other words \(y=1/u\). The derivative of this w.r.t. \(u\) can be calculated using the power rule: \(d y / d u = - u^{-2}\).

Step 2: Find \(d u / d x\)

We're given \(u=x^{3}+2 x^{2}\). The derivative of this w.r.t. \(x\) can be calculated by applying power rule to both terms separately: \(d u / d x = 3 x^{2} + 4x\).

Step 3: Find \(d y / d x\)

Now that we have both \(d y / d u\) and \(d u / d x\), we can use these to calculate \(d y / d x\) using chain rule. The chain rule states that \(d y / d x = (d y / d u) * (d u / d x)\). Thus we have \(d y / d x = (- u^{-2}) * (3 x^{2} + 4x)\). Now substitute \(u=x^{3}+2 x^{2}\) to get the final expression of \(d y / d x\) as \(- (x^{3}+2 x^{2})^{-2} * (3 x^{2} + 4x)\).

Key Concepts

Power Rule

The power rule is a fundamental technique in differentiation that significantly simplifies finding derivatives of functions where variables are raised to a power. If you have a function of the form \( x^n \), where \( n \) is any real number, then the power rule states that the derivative of this function is \( nx^{n-1} \). This means you bring the power down in front as a coefficient and subtract one from the original power.

Let's take a look at how the power rule is used in our original exercise. For Step 1, we find \( d y / d u \). We're given \( y = u^{-1} \), which is equivalent to \( y = u^{-1} \). By applying the power rule, the derivative becomes \( d y / d u = -u^{-2} \). The \(-1\) moves to the front as a multiplier, and the power of \(-2\) is the result of subtracting 1 from the original power of \(-1\).

In Step 2, we use the power rule to differentiate \( u = x^3 + 2x^2 \). The derivative \( d u / d x \) is found by applying the power rule to each term separately:

• The derivative of \( x^3 \) is \( 3x^2 \).
• The derivative of \( 2x^2 \) is \( 4x \).

Putting it all together, we get \( d u / d x = 3x^2 + 4x \).

This rule is handy because it allows you to quickly and efficiently find derivatives of polynomial functions with multiple terms.

Chain Rule

The chain rule is a critical tool in calculus for finding derivatives of composite functions, or functions within functions. It helps us "chain" together the derivatives of these nested functions.

In our exercise, Step 3 involves using the chain rule to find \( d y / d x \). The chain rule states that if you have a composite function \( y = f(g(x)) \), the derivative \( d y / d x \) is given by \( d y / d x = (d y / d u) \cdot (d u / d x) \).

Here, \( y = u^{-1} \) and \( u = x^3 + 2x^2 \). We've already found \( d y / d u = -u^{-2} \) and \( d u / d x = 3x^2 + 4x \). Using the chain rule, we calculate:

• \( d y / d x = -u^{-2} \cdot (3x^2 + 4x) \)

To express \( d y / d x \) in terms of \( x \), we substitute \( u = x^3 + 2x^2 \) back into the equation:

• \( d y / d x = - (x^3 + 2x^2)^{-2} \cdot (3x^2 + 4x) \)

The chain rule is essential because it allows us to differentiate more complex functions by breaking them down into simpler parts.

Derivative Calculation

Derivative calculation is the process of determining the rate of change of a function in relation to its variable. Calculus uses derivatives to understand how a function behaves, including its slope and inflection points.

In solving the exercise, we performed multiple derivative calculations. Each was important in understanding the overall rate of change between \( y \) and \( x \) through an intermediary variable \( u \).

Steps involved in the calculation helped us determine:

• \( d y / d u \): the rate at which \( y \) changes as \( u \) changes.
• \( d u / d x \): the rate at which \( u \) changes as \( x \) changes.
• \( d y / d x \): the overall rate of change of \( y \) with respect to \( x \), found by connecting the first two derivatives through the chain rule.

Derivative calculations empower us to predict and analyze the behavior of functions, informing decisions across physics, engineering, economics, and other fields by quantifying changes.