Problem 17 - Calculus an Applied Approach

Question

find the third derivative of the function. $$ f(x)=x^{5}-3 x^{4} $$

Step-by-Step Solution

Answer

The third derivative of the function \(f(x)=x^{5}-3 x^{4}\) is \(f'''(x) = 60x^{2}-72x\).

Step 1: Differentiation - First time

Applying the power rule to each term, the derivative of \(x^{5}\) is \(5 \cdot x^{5-1} = 5x^{4}\) and the derivative of \(-3x^{4}\) is \(-3 \cdot 4 \cdot x^{4-1} = -12x^{3}\). So, the first derivative function \(f'(x)\) is \(5x^{4}-12x^{3}\).

Step 2: Differentiation - Second time

Applying the power rule again to each term of \(f'(x)\), the derivative of \(5x^{4}\) is \(5 \cdot 4 \cdot x^{4-1} = 20x^{3}\) and the derivative of \(-12x^{3}\) is \(-12 \cdot 3 \cdot x^{3-1} = -36x^{2}\). So, the second derivative function \(f''(x)\) is \(20x^{3}-36x^{2}\).

Step 3: Differentiation - Third time

Applying the power rule again to each term of \(f''(x)\), the derivative of \(20x^{3}\) is \(20 \cdot 3 \cdot x^{3-1} = 60x^{2}\) and the derivative of \(-36x^{2}\) is \(-36 \cdot 2 \cdot x^{2-1} = -72x\). So, the third derivative function \(f'''(x)\) is \(60x^{2}-72x\).

Key Concepts

Power Rule

The Power Rule is a fundamental tool in calculus for finding the derivative of polynomial functions. It states that if you have a function of the form \(f(x) = ax^n\), where \(a\) is a constant and \(n\) is a real number, then the derivative of this function, denoted as \(f'(x)\), can be found using the formula:

• \(f'(x) = n \cdot ax^{n-1}\)

This means you multiply the coefficient \(a\) by the exponent \(n\), then decrease the exponent by 1.
For example, applying the Power Rule to \(x^5\) results in \(5x^4\), and applying it to \(-3x^4\) gives \(-12x^3\). This simplifies the process of differentiation, allowing us to find derivatives of polynomial terms efficiently.
In the exercise, each derivative step applies this rule to each term of the polynomial expression, moving from the first derivative to the third.

Differentiation

Differentiation is the process of finding a derivative. Derivatives represent the rate of change of a function as its input changes. For any polynomial, differentiation can be systematically performed using rules like the Power Rule.
The exercise asks us to find the third derivative of a polynomial function. To achieve this, we start with the original function, take its derivative to get the first derivative, differentiate again for the second, and one more time for the third.

• Start with the function \(f(x) = x^5 - 3x^4\).
• Apply differentiation to each term for the first derivative: \(f'(x) = 5x^4 - 12x^3\).
• Differentiate again: the second derivative \(f''(x) = 20x^3 - 36x^2\).
• Finally, differentiate to get the third derivative: \(f'''(x) = 60x^2 - 72x\).

Observing the simplification at each step helps in understanding how derivatives provide a simpler form representing the rate of change at each degree.

Calculus

Calculus is a branch of mathematics that studies continuous change and is broadly divided into two areas: differentiation and integration. Differentiation focuses on how functions change, using derivatives to provide precise measures of change.
This exercise falls under the realm of calculus by dealing with derivatives. By applying the Power Rule repeatedly, we explore how the function \(f(x) = x^5 - 3x^4\) evolves through multiple levels of differentiation.

• The first derivative gives the slope of the tangent line to the curve at any point.
• The second derivative provides insights on concavity, indicating whether the curve is bending upwards or downwards.
• The third derivative offers a deeper layer, sometimes related to acceleration in physical models.

These derivatives are valuable in multiple fields, from physics to economics, helping to solve complex real-world problems by modeling changes succinctly.