Question

Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=3 x^{3}-4 x$$

Short answer

Question: Given the function \(f(x) = 3x^3 - 4x\), determine the relationship between a small change in \(x\) (\(dx\)) and the corresponding change in \(y\) (\(dy\)). Answer: \(dy = (9x^2 - 4) dx\)

Step-by-step solution

Determine the Derivative of the Function

First, let's find the derivative of the function \(f(x) = 3x^3 - 4x\) with respect to \(x\). To do so, we'll use the power rule: \(d(ax^n)/dx = nax^{n-1}\). $$ \frac{d}{dx}(3x^3 - 4x) = \frac{d}{dx}(3x^3) - \frac{d}{dx}(4x) $$ Now applying the power rule to each term: $$ 3 \cdot \frac{d}{dx}(x^3) - 4 \cdot \frac{d}{dx}(x) = 3 (3x^2) - 4(1) $$ So the derivative of the function \(f(x)\) with respect to \(x\) is: $$ f'(x) = 9x^2 - 4 $$

Express the Relationship Between the Small Change in \(x\) and the Resulting Change in \(y\)

Now that we have the derivative of the function, we can express the relationship between the small change in \(x\) (\(dx\)) and the corresponding change in \(y\) (\(dy\)) as \(dy = f'(x)dx\). Replacing \(f'(x)\) with the derivative we found in step 1, we have: $$ dy = (9x^2 - 4) dx $$ So the relationship between a small change in \(x\) and the resulting change in \(y\) for the given function is: $$ dy = (9x^2 - 4) dx $$

Key concepts

Power Rule

The power rule is one of the simplest yet most important rules in calculus for finding derivatives. It's like a handy shortcut for differentiating polynomial functions. The power rule states that if you have a function of the form \( ax^n \), where \( a \) is a constant and \( n \) is a power on \( x \), then the derivative is \( nax^{n-1} \).
This means that you multiply the power \( n \) by the coefficient \( a \), and then you reduce the power by one.

Let’s see this in action with an example. Consider \( f(x) = 3x^3 \):

• You take the power, which is 3, and multiply it by the coefficient 3, to get 9.
• Then, reduce the power by one, changing \( x^3 \) to \( x^2 \).

So, the derivative of \( 3x^3 \) is \( 9x^2 \). Similarly, for the term \( -4x \), using the power rule means the derivative is simply -4, because \( x^1 \) becomes \( x^0 \) which is 1.

Differentiation

Differentiation is the process of finding the derivative of a function. When we differentiate a function, we are essentially finding how that function changes at any given point. Imagine you are driving a car; the speedometer tells you the speed at each moment. Differentiation is like the mechanism calculating that speed.

In mathematical terms, differentiation helps us understand how a small change in the input (like \( x \) in our function) causes a change in the output (\( y \)). It’s about measuring rates of change and is used extensively in areas ranging from physics to economics.

To differentiate the function \( f(x) = 3x^3 - 4x \), we use the power rule to get the derivative \( f'(x) = 9x^2 - 4 \). This derivative tells us the rate of change of \( y \) with respect to \( x \) for every value of \( x \).

Derivative Function

The derivative function, often denoted as \( f'(x) \) or \( \, dy/dx \), provides us with a new function. This function gives the slope of the tangent line to the graph of the original function at any given point. In our example, the original function is \( f(x) = 3x^3 - 4x \), and its derivative derived through differentiation is \( f'(x) = 9x^2 - 4 \).

The derivative function is a powerful tool because it allows us to compute the instantaneous rate of change.

• If \( f'(x) > 0 \), the function \( f(x) \) is increasing at \( x \).
• If \( f'(x) < 0 \), the function \( f(x) \) is decreasing at \( x \).

This can be exceptionally useful for identifying maximum and minimum points on the graph of the function, and understanding the behavior of the function over different intervals.

Change in Variables

When we talk about a small change in a variable in calculus, we often use the symbols \( dx \) and \( dy \). These represent tiny increments in \( x \) and \( y \), respectively. The expression \( dy = f'(x)dx \) is a differential equation that relates these small changes.
For our function \( f(x) = 3x^3 - 4x \), after finding the derivative \( f'(x) = 9x^2 - 4 \), we can express the small change relationship as \( dy = (9x^2 - 4)dx \).

This tells us that for a small change in \( x \), the resulting small change in \( y \) is determined by multiplying \( dx \) by the derivative \( f'(x) \). Understanding this relationship is crucial when analyzing how variations in input affect outcomes, especially in dynamic systems where conditions change continuously.