Question

Given \(y=u^{3}+2 u,\) where \(u=5-x^{2},\) find \(d y / d x\) by the chain rule.

Short answer

The derivative \( \frac{dy}{dx} = -154x + 60x^3 - 6x^5 \).

Step-by-step solution

Identify the Functions

Let's consider what we have: the function given is a composition of two components.- An outer function: \(y = u^3 + 2u\).- An inner function: \(u = 5 - x^2\).The task is to find \(\frac{dy}{dx}\), so we will need to find \(\frac{dy}{du}\) and \(\frac{du}{dx}\) first.

Differentiate the Outer Function

Differentiate \(y = u^3 + 2u\) with respect to \(u\):\[ \frac{dy}{du} = 3u^2 + 2 \]

Differentiate the Inner Function

Differentiate \(u = 5 - x^2\) with respect to \(x\):\[ \frac{du}{dx} = -2x \]

Apply the Chain Rule

To find \(\frac{dy}{dx}\), apply the chain rule:\[ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \]Substitute the results from the previous steps:\[ \frac{dy}{dx} = (3u^2 + 2)(-2x) \]

Substitute u back into the Expression

We have \(u = 5 - x^2\). Substitute this back into the expression obtained for \(\frac{dy}{dx}\):\[ \frac{dy}{dx} = (3(5-x^2)^2 + 2)(-2x) \]

Simplify the Expression

Now expand and simplify the expression:First, find \((5-x^2)^2\):\((5-x^2)^2 = 25 - 10x^2 + x^4\).Substitute back:\[ \frac{dy}{dx} = (3(25 - 10x^2 + x^4) + 2)(-2x) \]Simplify inside the parentheses:\[ 3(25 - 10x^2 + x^4) + 2 = 75 - 30x^2 + 3x^4 + 2 \]Further simplification:\[ 77 - 30x^2 + 3x^4 \].Substitute back:\[ \frac{dy}{dx} = (77 - 30x^2 + 3x^4)(-2x) \]

Final Expression

Finally, distribute \((-2x)\) to evaluate \(\frac{dy}{dx}\):\[ \frac{dy}{dx} = -2x(77 - 30x^2 + 3x^4) = -2x \cdot 77 + 2x \cdot 30x^2 - 2x \cdot 3x^4 \].This results in:\[ -154x + 60x^3 - 6x^5 \].

Key concepts

Differentiation

Differentiation is a crucial concept in calculus that involves finding the derivative of a function. This derivative can provide insights into how a function changes as its input changes. In this exercise, we are given a composite function and asked to find the derivative \( \frac{dy}{dx} \). Since the function involves a composition, we utilize the chain rule for differentiation.

The chain rule is essential when dealing with composite functions. It helps us relate the rate of change of an "outer function" to that of an "inner function." Essentially, we find the derivative of each function separately and then multiply them to get the complete rate of change. Understanding the process of differentiation using the chain rule allows students to solve complex problems involving functions within functions.

Outer Function

The outer function is the overarching function in a composition that applies to the result of the inner function. In this problem, the outer function is given as \( y = u^3 + 2u \). Our task is to find how this function changes with respect to the inner function, \( u \).

To differentiate the outer function, we first introduce the notation \( \frac{dy}{du} \), which reminds us that we are initially focusing on the variable \( u \). The differentiation process involves basic power and constant rule applications:

• The derivative of \( u^3 \) is \( 3u^2 \) - applying the power rule.
• The derivative of \( 2u \) is simply \( 2 \) - the constant rule.

Thus, the derivative of the outer function with respect to \( u \) is \( \frac{dy}{du} = 3u^2 + 2 \). Developing comfort with identifying and differentiating the outer function is a skill that greatly simplifies handling more complex problems.

Inner Function

The inner function refers to the component that is substituted into the outer function. In this case, we are dealing with the function \( u = 5 - x^2 \), and we are interested in how it changes with respect to \( x \).

Differentiating an inner function involves finding \( \frac{du}{dx} \). Here, the derivative of \( 5 \) is zero because it is a constant, while the derivative of \( -x^2 \) is \( -2x \). So, we find that \( \frac{du}{dx} = -2x \).

Mastering the differentiation of inner functions allows us to understand how inputs transform before undergoing further operations in the outer function. Properly identifying the inner function and correctly applying basic differentiation rules is crucial in the application of the chain rule.

Simplification

Simplification is the final phase of processing an already differentiated expression, making it easier to understand or usable for further calculations. After finding \( \frac{dy}{dx} \) using the chain rule, the result may look complex or unwieldy.

In this problem, we applied the chain rule to get an expression involving both \(-2x\) and \((3u^2 + 2)\). The next step involves substituting the original expression for \( u \), \( 5 - x^2 \), back into our derived function. Expanding the squared term and distributing it allows us to simplify further:

• First, compute \((5-x^2)^2 = 25 - 10x^2 + x^4\).
• Then substitute back into the equation.
• Finally, distribute \(-2x\) over the simplified polynomial.

Through expansion and distribution, we eventually arrived at the simplified form \( -154x + 60x^3 - 6x^5 \). Simplification ensures the result is clean and interpretable, allowing for meaningful use in applications or further mathematical procedures.