Problem 15 - Calculus and its applications

Question

Find \(f^{\prime \prime}(x)\) $$ f(x)=x^{1 / 5} $$

Step-by-Step Solution

Answer

\(f''(x) = -\frac{4}{25}x^{-9/5}\).

Step 1: Identify the Function

The function we are dealing with is \( f(x) = x^{1/5} \). Our task is to find the second derivative, \( f''(x) \).

Step 2: Differentiate to Find the First Derivative

Using the power rule for differentiation, which states \( \frac{d}{dx} x^n = nx^{n-1} \), let's differentiate \( f(x) = x^{1/5} \). The first derivative, \( f'(x) \), is \( \frac{1}{5}x^{-4/5} \).

Step 3: Differentiate Again to Get the Second Derivative

Now, differentiate the first derivative \( f'(x) = \frac{1}{5}x^{-4/5} \) once more using the power rule. This will give us the second derivative: \( f''(x) = \frac{1}{5} \cdot (-\frac{4}{5})x^{-9/5} = -\frac{4}{25}x^{-9/5} \).

Key Concepts

Differentiation

Differentiation is like the math toolkit we use to find out how things are changing. Imagine driving a car and watching the speedometer to see how fast you're going. Similarly, in calculus, differentiation lets us see how functions change at any given point.
Every function has its own unique rate of change, and by finding the derivative of a function, we get a new function that tells us the rate of change of the original function. For example, in the exercise, the function given is a power function:

• Original function: \( f(x) = x^{1/5} \)
• First derivative: \( f'(x) = \frac{1}{5}x^{-4/5} \)
• Second derivative: \( f''(x) = -\frac{4}{25}x^{-9/5} \)

By differentiating, we've transformed the original function to give us new insights on how it behaves and changes over its domain. Differentiation is a fundamental concept because it allows us to understand and predict the behavior of mathematical models.

Power Rule

The power rule is a straightforward and powerful tool in calculus to help find derivatives of power functions like \( x^n \). It's like a magic wand that simplifies differentiation. The rule states that if you have a function \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \).
Here's how it works in practice:

• Take the exponent of \( x \) and multiply the entire term by it.
• Reduce the original exponent by one to find the new exponent for \( x \) in the derivative.

In the example provided:

• Function: \( x^{1/5} \)
• Use the power rule: the first derivative becomes \( \frac{1}{5}x^{-4/5} \)
• If you differentiate again: apply the rule to get the second derivative \( -\frac{4}{25}x^{-9/5} \)

This rule is handy because it turns what could be a complex process into something quick and easy to apply, helping you calculate derivatives more efficiently in many math problems.

Second Derivative

The second derivative is like taking the pulse of how change itself is changing. It adds another layer of understanding to our analysis of functions. While the first derivative gives us the function's rate of change, the second derivative informs us about the rate of change of that rate – essentially, it's the ‘acceleration’ of the function.
In simpler terms:

• A positive second derivative suggests the original function is concave up, like a bowl.
• A negative second derivative indicates the function is concave down, like an upside-down bowl.

In our given exercise, we found the first derivative \( f'(x) = \frac{1}{5}x^{-4/5} \). By differentiating this once more, we calculated the second derivative: \( f''(x) = -\frac{4}{25}x^{-9/5} \). The negative sign of our second derivative tells us that the original function \( f(x) = x^{1/5} \) is concave down.Understanding the second derivative is crucial as it assists in analyzing critical points and aiding in curve sketching, providing us with insights into the behavior and shape of graphs.