Question

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{1}{3 x^{2}+1} $$

Short answer

The derivative is \( \frac{dy}{dx} = -\frac{6x}{(3x^2 + 1)^2} \).

Step-by-step solution

Identify the Differentiation Rule

The given function is of the form \( y = \frac{1}{u} \), where \( u = 3x^2 + 1 \). We need to use the quotient rule or the chain rule to differentiate this type of function. For simplicity, we will use the chain rule here.

Differentiate the Inner Function

First, differentiate the inner function \( u = 3x^2 + 1 \) with respect to \( x \). The derivative is \( \frac{du}{dx} = 6x \).

Apply the Chain Rule

The derivative of \( y = \frac{1}{u} = u^{-1} \) with respect to \( u \) is \( \frac{dy}{du} = -u^{-2} \). Now, apply the chain rule: \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = -u^{-2} \times 6x \).

Substitute Back the Inner Function

Replace \( u \) with \( 3x^2 + 1 \) in the expression from Step 3 to get the derivative in terms of \( x \):\[\frac{dy}{dx} = -\frac{6x}{(3x^2 + 1)^2}.\]

Simplify the Expression

Ensure the expression is simplified, but in this case, it is already in its simplest form. There is no further simplification needed for \(-\frac{6x}{(3x^2 + 1)^2}\).

Key concepts

Chain Rule

The chain rule is a fundamental concept when dealing with composite functions. A composite function is when a function is inside another function. The chain rule is especially handy because it lets us differentiate such functions by breaking them down into simpler parts.
For instance, if you have a function of the form \( f(g(x)) \), the chain rule states that the derivative of this function is \( f'(g(x)) \cdot g'(x) \). It's basically "the derivative of the outer function times the derivative of the inner function."

• Take the derivative of the outer function, treating the inner function as a variable.
• Multiply by the derivative of the inner function itself.

In the original solution, we applied the chain rule to \( y = \frac{1}{u} = u^{-1} \). We found the derivative of the outer function, \( -u^{-2} \), and then multiplied it by the derivative of the inner function, \( 6x \). This is how the chain rule efficiently helps in differentiating composite functions.

Derivative

Derivatives are the cornerstone of calculus. They provide the slope of a function at any point, essentially describing how the function changes. Technically, the derivative of a function \( f(x) \) with respect to \( x \) is the limit of the average rate of change as the interval approaches zero. It's defined as:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]
The process of finding a derivative is known as differentiation. We differentiate to find rates of change, such as velocity or acceleration, in various contexts.

• The derivative of a constant is 0.
• The derivative of \( x^n \) is \( nx^{n-1} \).

In the exercise, the derivative of the inner function \( u = 3x^2 + 1 \) was calculated as \( 6x \). Derivatives help solve a variety of real-world problems by modeling the change of quantities.

Quotient Rule

The quotient rule is another essential technique for differentiating functions, specifically rational functions where one function divides another. The quotient rule formula is:
\[ \left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2} \]
This rule is particularly useful when directly dealing with expressions like \( \frac{f(x)}{g(x)} \). Here's the breakdown:

• \( f \) and \( g \) are functions of \( x \).
• \( f' \) is the derivative of the numerator function \( f \).
• \( g' \) is the derivative of the denominator function \( g \).
• Combine the derivatives as per the formula.

Although the chain rule was used in the solution for simplicity, the quotient rule could also be applied for this exercise, aligning with the same end result. When dealing with simple compositions, it's consistently crucial to identify the simpler path to the answer.