Problem 11 - Calculus an Applied Approach

Question

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

Step-by-Step Solution

Answer

\(dy/du = 1/2 * u^{-1/2}\), \(du/dx = -2x\), and \(dy/dx = -x * (3 - x^2)^{-1/2}\).

Step 1: Derive \(dy/du\)

First, find the derivative of y with respect to u. The function y is given as \(y = \sqrt{u}\). We write this in the form of power as \(y = u^{1/2}\). Now, if we differentiate this with respect to u, we get using the power rule (\(dy/du = n*x^{(n-1)}\)) for \(dy/du = \frac{1}{2} * u^{-1/2}\).

Step 2: Derive \(du/dx\)

Now, we find the derivative of u with respect to x. The function u is given as \(u = 3 - x^2\). Differentiating this with respect to x, we get for \(du/dx = -2x\).

Step 3: Derive \(dy/dx\)

Lastly, we find the derivative of y with respect to x by using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Here, y is a function of u, and u is a function of x. Therefore, to find \(dy/dx \), we multiply \(dy/du\) and \(du/dx\) which gives \(dy/dx = (-2x * 1/2 * u^{-1/2})\). By substituting \(u = 3 - x^2\) in the above expression, we get \(dy/dx = [-x * (3 - x^2)^{-1/2}]\).

Key Concepts

Derivative of a Function

Understanding the derivative of a function is essential in calculus, as it describes how a function changes with respect to a variable. When we say derivative, we're referring to the rate of change or the slope of the function curve at a particular point.

When solving a problem like finding the derivative of a composite function, it's important to first identify the individual functions involved. These functions can be simple or complex, such as polynomials or radical expressions. In the exercise above, we encountered two functions: \(y = \sqrt{u}\) and \(u = 3 - x^2\).

Calculating the derivative involves computing the rate of change concerning different variables. For instance, when we determined \(dy/du\), we found how \(y\) changes with \(u\), and when finding \(du/dx\), we calculated how \(u\) changes with \(x\). This is the foundational process of differentiation necessary for finding the overall derivative \(dy/dx\), especially in composite functions.

Power Rule

The power rule is one of the most straightforward rules of differentiation, applicable when dealing with expressions of the form \(x^n\), where \(n\) is any real number.

For our context, the power rule is given by \(d/dx \, (x^n) = n \cdot x^{n-1}\). This means you take the exponent \(n\), bring it down as a coefficient, and then reduce the exponent by one.

In the exercise, the function \(y = \sqrt{u}\) was transformed into \(y = u^{1/2}\). Applying the power rule involves the following steps:

• First, identify \(n\) in the expression. Here, \(n = 1/2\).
• Multiply \(n\) by the variable raised to the power of \(n-1\), resulting in \(dy/du = \frac{1}{2} \cdot u^{-1/2}\).

This calculation helps find the rate at which \(y\) changes with respect to \(u\). It simplifies functions, making differentiation a straightforward process.

Composite Function Differentiation

Composite functions combine two or more functions, where the output of one becomes the input of another. In calculus, we use the chain rule to differentiate such functions.

The chain rule states that to differentiate a composite function, you must multiply the derivative of the outer function by the derivative of the inner function. Mathematically, it can be written as \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

In the original exercise, finding \(dy/dx\) required us first to identify the outer and inner functions: \(y = \sqrt{u}\) as the outer function and \(u = 3 - x^2\) as the inner function. Here’s a simplified breakdown:

• First, we found \(dy/du = \frac{1}{2} \cdot u^{-1/2}\) using the power rule on the outer function \(y\).
• Next, we found \(du/dx = -2x\) by differentiating the inner function \(u = 3 - x^2\).
• Then, we multiplied these derivatives as per the chain rule: \(dy/dx = (-2x \cdot \frac{1}{2} \cdot u^{-1/2})\).

Finally, substituting \(u = 3 - x^2\) back into the expression, we simplified to get the final result: \(-x \cdot (3 - x^2)^{-1/2}\). This approach highlights how interconnected derivatives are in composite functions.