Problem 19 - Calculus Single Variable

Question

Calculate \(f^{\prime}(x),\) and sketch the graph of the equation \(y=f^{\prime}(x)\). $$ f(x)=3-2 x^{3} $$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = -6x^2\), a downward-opening parabola.

Step 1: Identify the Function

Given the function is \(f(x) = 3 - 2x^3\). We need to find its derivative, which is \(f'(x)\).

Step 2: Apply the Power Rule

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Apply this to each term in \(f(x)\).

Step 3: Differentiate Each Term

Differentiate the constant term 3, which gives 0, since the derivative of a constant is 0. Differentiate \(-2x^3\) using the power rule: \(-2(3)x^{3-1} = -6x^2\).

Step 4: Combine the Derivatives

Combine the results of the differentiation: \(f'(x) = 0 - 6x^2 = -6x^2\).

Step 5: Sketch the Graph of the Derivative

The function \(f'(x) = -6x^2\) is a downward-opening parabola (because of the negative coefficient) with its vertex at the origin, which means the graph is obtained by reflecting the graph of \(y = 6x^2\) downward across the x-axis.

Key Concepts

Power Rule

To understand derivatives, one of the first rules you should master is the power rule. This rule is a simple but powerful tool for differentiating polynomial functions. In essence, the power rule helps us find the derivative of a term in the form of \(x^n\), where \(n\) is any real number. The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\).
This means you simply multiply the exponent \(n\) by the coefficient of the term and then subtract one from the exponent.

• Example: For \(f(x) = x^4\), the derivative \(f'(x)\) would be \(4x^{3}\).
• If there is a coefficient before the \(x\), like \(5x^4\), the derivative would be \(20x^3\).

The power rule simplifies the process of differentiation and is a foundation for exploring more complex functions.

Polynomial Functions

Polynomials are mathematical expressions that include terms of variables raised to powers, such as \(x^3\) or \(2x^5\). These functions can be simple, like \(x^2\), or complex involving multiple terms. Polynomials are essential in calculus because they are smooth, continuous, and easy to differentiate.
When differentiating a polynomial function, apply the power rule to each term separately and sum the results.

• A polynomial like \(f(x) = 3 - 2x^3\) involves a constant term and a power term.
• To differentiate, apply the power rule and consider each term individually.

The result is also a polynomial, in our case, \(f'(x) = -6x^2\), which retains the smooth properties of the original function.

Derivative of Constants

When it comes to derivatives, constants play a special role. A constant is a number without any variable associated with it and does not change. When you take the derivative of a constant, the result is always zero.
This is because derivatives represent the rate of change, and a constant does not change, hence its rate of change is zero.
For example, if you have \(f(x) = 7\), the derivative \(f'(x)\) is 0. Similarly, in the given function \(f(x) = 3 - 2x^3\), the term 3 is a constant, so its derivative is 0.

• This simplifies the differentiation process since constants can be disregarded when calculating the derivative of a function with multiple terms.

Understanding the derivative of constants is crucial for accurately applying rules of differentiation, especially in polynomial functions.