Problem 32 - Technical Mathematics with Calculus

Question

Write the differential \(d y\) for each function. $$y=\left(2-3 x^{2}\right)^{3}$$

Step-by-Step Solution

Answer

The differential of the function is given by \(dy = -18x(2 - 3x^2)^2 dx\).

Step 1: Use the Chain Rule

To find the differential of the given function, one must apply the chain rule, which, in this context, states that if a variable y depends on u which itself depends on x, then the differential dy can be found as the product of the derivative of y with respect to u and the differential of u. Here, let u = (2 - 3x^2).

Step 2: Differentiate the Outer Function

With u defined, y = u^3, so the derivative of y with respect to u is dy/du = 3u^2. So, dy = 3u^2 du.

Step 3: Differentiate the Inner Function

Next, differentiate u with respect to x. The function u = 2 - 3x^2 is a simple polynomial, and its derivative is du/dx = -6x. Hence, du = -6x dx.

Step 4: Combine the Differentiated Parts

Finally, combine the results of steps 2 and 3 to find dy. Using the chain rule, dy = 3u^2(-6x dx) = -18xu^2 dx.

Step 5: Substitute back for u

Replace u with the original expression in terms of x to find the complete differential of y. So, dy = -18x(2 - 3x^2)^2 dx.

Key Concepts

Chain Rule

The chain rule is a fundamental principle in differential calculus. It comes into play when you're dealing with composite functions, where one function is nested within another. Think of it as peeling an onion, layer by layer, to differentiate each function in turn.

For a function like \( y = (2 - 3x^2)^3 \), the chain rule allows us to find the derivative by breaking down the process into simpler steps. We consider \( y \) as a function of \( u \) where \( u \) is a function of \( x \). Here's how the chain rule is applied:

• Identify the inner function, \( u \), and the outer function \( y(u) \).
• Differentiate the outer function with respect to \( u \), treating \( u \) as the variable.
• Differentiate the inner function with respect to \( x \) to find \( \frac{du}{dx} \).
• Multiply the derivatives to get the derivative of the composite function with respect to \( x \).

This rule is essential because it allows us to handle complex derivatives that would otherwise be difficult to manage.

Differential dy

In the realm of calculus, the differential \( dy \) represents the change in the function \( y \) as \( x \) changes by an infinitesimally small amount, \( dx \). This concept is key to understanding rates of change and slopes of curves at specific points.

When we use the chain rule to differentiate a composite function, we calculate \( dy \) by considering it as the product of two differentials: the differential of the outer function and the differential of the inner function.

• Firstly, we find \( \frac{dy}{du} \) and multiply it by the infinitesimal \( du \) to get the differential \( dy \).
• Secondly, we differentiate the inner function to find \( \frac{du}{dx} \), and then multiply by \( dx \) to get \( du \).
• The product of \( \frac{dy}{du} \) and \( \frac{du}{dx} \) gives us \( \frac{dy}{dx} \), and their differentials give us \( dy \).

It's like a domino effect, where the change in one variable causes a ripple effect, ultimately leading to the change in our function \( y \). By finding \( dy \), we obtain a precise expression for this change.

Polynomial Differentiation

Differentiating polynomials is usually one of the first applications of derivatives that students encounter. The process is straightforward, thanks to the power rule, which states that for any polynomial term \( ax^n \), the derivative is \( anx^{n-1} \).

Let's consider the polynomial \( u = 2 - 3x^2 \). Applying the power rule to each term separately, we get the derivative \( \frac{du}{dx} = -6x \).

• For the constant term 2, the derivative is 0 because the rate of change of a constant term with respect to \( x \) is zero.
• The term \( -3x^2 \) is differentiated as \( -3 \) times the derivative of \( x^2 \) with respect to \( x \), which is \( 2x \), so we get \( -6x \).

Polynomial differentiation is a core skill in calculus because polynomials are so common. By knowing the power rule, one can efficiently find derivatives for these types of functions without needing to resort to complex rules or formulas.