Problem 78 - Calculus for Biology and Medicine

Question

Find the first and the second derivatives of each function. \(f(x)=\frac{1}{x^{2}}+x-x^{3}\)

Step-by-Step Solution

Answer

The first derivative is \( f'(x) = -2x^{-3} + 1 - 3x^2 \) and the second derivative is \( f''(x) = 6x^{-4} - 6x \).

Step 1: Rewrite the Function

Before differentiating, rewrite the function in a more convenient form: \[ f(x) = x^{-2} + x - x^3 \]. This allows us to apply power rule directly for differentiation.

Step 2: Differentiate to Find the First Derivative

Apply the power rule \( \frac{d}{dx} x^n = n x^{n-1} \) to each term in \( f(x) = x^{-2} + x - x^3 \):- The derivative of \( x^{-2} \) is \( -2x^{-3} \).- The derivative of \( x \) is \( 1 \).- The derivative of \( -x^3 \) is \( -3x^2 \).Thus, the first derivative is:\[ f'(x) = -2x^{-3} + 1 - 3x^2 \].

Step 3: Differentiate Again to Find the Second Derivative

Differentiate \( f'(x) = -2x^{-3} + 1 - 3x^2 \) using the power rule again:- The derivative of \( -2x^{-3} \) is \( 6x^{-4} \).- The derivative of \( 1 \) is \( 0 \).- The derivative of \( -3x^2 \) is \( -6x \).Therefore, the second derivative is:\[ f''(x) = 6x^{-4} - 6x \].

Key Concepts

Understanding the Power Rule

The power rule is a fundamental tool for calculating derivatives in calculus. It is particularly useful for functions expressed as powers of a variable. The rule itself states that if you have a function of the form \( f(x) = x^n \), then its derivative can be found using the formula \( \frac{d}{dx} x^n = nx^{n-1} \). This concept makes it incredibly easy to differentiate terms that involve powers of \( x \).

In our exercise, we first rewrote the function \( f(x) = \frac{1}{x^2} + x - x^3 \) in terms of negative and positive powers: \( f(x) = x^{-2} + x - x^3 \). This rewriting helps utilize the power rule effectively for each term:

• For \( x^{-2} \), applying the power rule gives the derivative \( -2x^{-3} \).
• For \( x \), which is really \( x^1 \), the derivative is simply \( 1 \times x^{0} = 1 \).
• For \( -x^3 \), we obtain \( -3x^2 \) as the derivative.

This simplification makes taking derivatives straightforward and reduces the risk of error.

Finding the First Derivative

The first derivative of a function measures how the function's output changes as the input changes. That is, it tells us the rate of change or the slope of the function at any given point. To find the first derivative, we apply the power rule to each term of our rewritten function:

- \( x^{-2} \) becomes \( -2x^{-3} \)
- \( x \) becomes \( 1 \)
- \( -x^3 \) becomes \( -3x^2 \)

Adding these, the first derivative of our function \( f(x) = x^{-2} + x - x^3 \) is:
\[ f'(x) = -2x^{-3} + 1 - 3x^2 \]

This result provides a new function that describes the slope of the original function at every point along its graph. Understanding the first derivative is crucial as it helps to reveal where the function is increasing, decreasing, or has any constant segments.

Exploring the Second Derivative

The second derivative gives us a deeper understanding of the function's behavior by showing us the rate at which the slope is changing. In practical terms, it's the derivative of the derivative. It can reveal whether a function is concave or convex, or identify points of inflection.

To find the second derivative, we differentiate the first derivative \( f'(x) = -2x^{-3} + 1 - 3x^2 \) again. Using the power rule once more for each term results in:

• The derivative of \( -2x^{-3} \) is \( 6x^{-4} \).
• The derivative of the constant \( 1 \) is \( 0 \), because constants do not change as \( x \) changes.
• The derivative of \( -3x^2 \) is \( -6x \).

Thus, the second derivative is:
\[ f''(x) = 6x^{-4} - 6x \]
This second derivative informs us about the concavity of the original function \( f(x) \). If the second derivative is positive, the function is concave up (like a cup); if negative, concave down (like a frown). It helps in sketching curves and analyzing how a function behaves beyond just slope changes.