Problem 18 - Calculus and its applications

Question

Find \(f^{\prime \prime}(x)\) $$ f(x)=4 x^{-3} $$

Step-by-Step Solution

Answer

The second derivative is \( f''(x) = 48x^{-5} \).

Step 1: Identify the Function

We are given the function \( f(x) = 4x^{-3} \). Our task is to find the second derivative of this function, \( f''(x) \).

Step 2: Find the First Derivative

To find \( f'(x) \), differentiate \( f(x) = 4x^{-3} \) with respect to \( x \). Use the power rule \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \), thus: \( f'(x) = 4 \cdot (-3) \cdot x^{-3-1} = -12x^{-4} \).

Step 3: Find the Second Derivative

Now, differentiate \( f'(x) = -12x^{-4} \) to find \( f''(x) \). Using the power rule again: \( f''(x) = -12 \cdot (-4) \cdot x^{-4-1} = 48x^{-5} \).

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus for differentiating functions of the form \(x^n\), where \(n\) is any real number. This rule is particularly handy since it simplifies the differentiation process considerably. The power rule states that if you have a function \(f(x) = x^n\), its derivative is \(f'(x) = n \cdot x^{n-1}\). This means you bring the exponent \(n\) down as a coefficient and subtract one from the original exponent.

• Effective for both positive and negative integers.
• Applies to fractional and irrational exponents too.
• Simplifies differentiation by avoiding limits each time.

For our exercise, the function \(f(x) = 4x^{-3}\) was handled using this rule. We first took the derivative to get \(f'(x) = -12x^{-4}\), and applied the rule a second time to find \(f''(x) = 48x^{-5}\). The power rule makes these steps straightforward.

Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function's output changes with respect to changes in the input. This is useful for understanding rates of change, like velocity in physics or slope of curves in mathematics. The derivative of a function is often denoted by \(f'(x)\) or \(\frac{dy}{dx}\).

• Allows prediction of how functions behave.
• Provides insight into optimization problems.
• Helps in approximating functions using linear models.

In the given exercise, differentiation was used to find both the first and second derivatives. Starting with \(f(x) = 4x^{-3}\), we found the first derivative to be \(f'(x) = -12x^{-4}\), and then proceeded to obtain the second derivative as \(f''(x) = 48x^{-5}\). Differentiation helped determine how this function changes over time.

Functions

Functions are one of the core building blocks in mathematics, used to describe relationships between sets of numbers or objects. They're expressed in the form \(f(x)\), where \(x\) represents the function's input and \(f(x)\) is the output.Functions can be as simple as \(y = x + 2\), or as complex as trigonometric or logarithmic functions.They are used to model real-world phenomena such as growth patterns, financial markets, and even population dynamics.

• Convert inputs into outputs through a defined rule.
• Can be manipulated and analyzed using calculus techniques like differentiation.
• Essential for modeling in various scientific and engineering disciplines.

In the exercise, the function \(f(x) = 4x^{-3}\) was examined. This function's behavior was analyzed by differentiating it twice, showing how it changes with respect to \(x\). Understanding functions and their derivatives allows deeper insights into complex behavior.