Question

In Exercises, find the third derivative of the function. $$ f(x)=x^{4}-2 x^{3} $$

Short answer

The third derivative of the function \(f(x) = x^4 - 2x^3\) is \(f'''(x) = 24x - 12\)

Step-by-step solution

Find the First Derivative

Using the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\), find the first derivative of the function. Therefore, \(f'(x) = 4x^{3} - 6x^{2} \)

Find the Second Derivative

To find the second derivative, we're going to differentiate the first derivative. Applying the power rule again, we get \(f''(x) = 12x^{2} - 12x \)

Find the Third Derivative

The third derivative is the derivative of the second derivative. Applying the power rule one more time gives us \( f'''(x) = 24x - 12 \)

Key concepts

Power Rule

The power rule is one of the most fundamental tools in calculus, particularly useful for finding derivatives of polynomial functions. It helps us understand how the rate of change of a function behaves. When applying the power rule, we take a function of the form \(x^n\) and transform it to find its derivative. The rule states: if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).
This means you multiply the exponent \(n\) by the base and reduce the original exponent by one. Let's consider an example: for the function \(x^4\), the derivative according to the power rule would be \(4x^3\). Similarly, \(x^3\) would be differentiated to \(3x^2\).
This rule significantly simplifies the process of differentiation and is particularly powerful when dealing with polynomial expressions, as it allows for rapid transformation of terms one by one.

Derivatives

Derivatives are a core concept in calculus, representing the rate of change of a function. They are essentially the slope of the function at any given point and are crucial for understanding dynamics in various fields such as physics, biology, and economics. Differentiation is the process of finding a derivative.

• The first derivative of a function \(f(x)\) provides information on the slope of the tangent line at any point \(x\).
• The second derivative, \(f''(x)\), gives us insight into the concavity of the function—whether it curves upwards or downwards.
• Higher-order derivatives, such as the third derivative, can describe more intricate changes like the rate of change of the concavity.

The step-by-step manner in which derivatives are calculated ensures precision and aids in pinpointing how functions behave across different intervals. By starting with simple operations like applying the power rule, we can build up to complex equations, progressively revealing layers of change within the function.

Third Derivative

The third derivative of a function represents a deeper level of analyzing change, often related to the smoothness or jerk in physical terms. While the first derivative is about the immediate rate of change, the second derivative indicates acceleration, and the third derivative often describes the "jerk" in mechanics, or the rate of change of acceleration.
To find the third derivative, you differentiate the second derivative. For instance, if you have \(f''(x) = 12x^2 - 12x\), applying the power rule gives you \(f'''(x) = 24x - 12\).
This step is similar to the first and second derivative in its reliance on the power rule but defines how the acceleration itself is changing, which can be particularly valuable in fields that model motion or growth trends. This layer of differentiation helps in understanding the subtleties of change beyond simple acceleration.