Problem 33 - Precalculus with Limits

Question

In Exercises 29-42, find the derivative of the function. \(f(x) = 4-3x^2\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = 4-3x^2\) is \(-6x\).

Step 1: Identify the function

The function given is \(f(x) = 4-3x^2\). This equation is a simple quadratic function, which can be broken down into two terms: a constant term (4), and a variable term (-3x^2). These terms will be differentiated separately.

Step 2: Take the derivative of the constant term

The derivative of any constant is always zero. For our function, the constant term is \(4\). So the derivative of this term is zero.

Step 3: Take the derivative of the variable term

The derivative of the variable term \(-3x^2\) is based on the power rule of differentiation. The power rule states that the derivative of \(x^n\), where \(n\) is any real number, is \(n*\) coefficient of \(x^{n-1}\). Here, \(n\) is \(2\) and the coefficient is \(-3\). So the derivative of this term is \(-3*2*x^{2-1} = -6x\).

Step 4: Combine the derivatives of each term

Now, add the derivatives of the two terms together. The total derivative of the function is the sum of the derivatives of its parts. So, the derivative of the function \(f(x) = 4-3x^2\) equals \(0 - 6x\), which simplifies to \(-6x\).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus used to determine the derivative of functions. It's a simple yet powerful technique, particularly useful when dealing with polynomial functions. The rule can be summarized as follows:

• If you have a term in the form of \(x^n\), where \(n\) is a real number, its derivative is \(n \times x^{n-1}\).

This means you multiply the term by the exponent, then reduce the exponent by one.
For example, for the term \(x^2\), applying the power rule: multiply the power \(2\) by the coefficient (even if it's not visible, it's \(1\)) and reduce the power by one:

• \(1 \times 2 \times x^{2-1} = 2x\).

In the context of our function \(f(x) = 4 - 3x^2\), applying the power rule to \(-3x^2\) involves:

• First, multiplying the power \(2\) by the coefficient \(-3\).
• Then, reducing the power by one: \(-3 \times 2 \times x^{2-1} = -6x\).

This gives the derivative \(-6x\).

Constant Term

A constant term in a function is a number on its own without any variables attached to it. This is an important feature because it simplifies the calculation of derivatives.

• In our example, the constant term is \(4\).

The derivative of any constant is always zero because a constant does not change, it has no rate of change.
To visualize this, consider a flat line on a graph corresponding to a constant value. This line has no slope, meaning its steepness or rate of change is zero.

• So, for the function \(f(x) = 4 - 3x^2\), the derivative of \(4\) is \(0\).

When finding the derivative of a function, the constant term drops out, simplifying the expression.

Quadratic Function

Quadratic functions are polynomial functions of degree 2, which means the highest power of the variable \(x\) is 2.

• The general form for a quadratic function is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

In the exercise, the given function is \(f(x) = 4 - 3x^2\).
This is a quadratic function simplified with the terms rearranged.

• The coefficient \(-3\) in front of \(x^2\) tells us the direction and steepness of the parabola if graphed.
• The constant term \(4\) shifts the function vertically on the graph.

When taking the derivative, apply the power rule to \(-3x^2\) for differentiation.
This process leaves the derivative as \(-6x\), providing the rate of change indicating the slope of the tangent to the parabola at any point \(x\). This derivative reflects the linear term resulting from differentiating the quadratic term.