Problem 1 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, find the first and second derivative of the function defined by the given equation. $$ f(x)=x^{5}-2 x^{3}+x $$

Step-by-Step Solution

Answer

The first derivative is \( f'(x) = 5x^4 - 6x^2 + 1 \) and the second derivative is \( f''(x) = 20x^3 - 12x\).

Step 1 - Find the first derivative

To find the first derivative of the function, use the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \). For the function \(f(x) = x^5 - 2x^3 + x \), differentiate each term separately. \ \ For \(x^5\), the derivative is \ \(5x^4\). \ \ For \( -2x^3 \), the derivative is \ \(-6x^2\). \ \ For \ \( x \), the derivative is \ \(1\) . Therefore, the first derivative is \ \(f'(x) = 5x^4 - 6x^2 + 1\).

Step 2 - Find the second derivative

Differentiate \(f'(x) = 5x^4 - 6x^2 + 1\) to find the second derivative. Using the power rule: \ \ For \ \( 5x^4 \), the derivative is \ \(20x^3\). \ \ For \ \( -6x^2 \), the derivative is \ \(-12x\).\ \ For \ \(1\) which is constant, the derivative is 0. Therefore, the second derivative is \ \(f''(x) = 20x^3 - 12x\).

Step 3: Summary

The first derivative of the function \( f(x) = x^5 - 2x^3 + x \) is \( f'(x) = 5x^4 - 6x^2 + 1 \) and the second derivative is \( f''(x) = 20x^3 - 12x \).

Key Concepts

First Derivative

The first derivative of a function measures the rate at which the function's value changes. Think of it as the function's slope at any given point. To find the first derivative of the function \( f(x) = x^5 - 2x^3 + x \), we use the power rule. The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). For each term in \( f(x) \), we apply this rule separately:

• For \( x^5 \), the derivative is \( 5x^4 \).
• For \( -2x^3 \), the derivative is \( -6x^2 \).
• For \( x \), the derivative is \( 1 \).

Putting it all together, the first derivative is \( f'(x) = 5x^4 - 6x^2 + 1 \). This tells us how the function \( f(x) \) changes with respect to \( x \).

Second Derivative

The second derivative represents how the first derivative changes as \( x \) changes. You can think of it as the acceleration of the function's value. To find the second derivative, we differentiate the first derivative \( f'(x) = 5x^4 - 6x^2 + 1 \). Again, we apply the power rule:

• For \( 5x^4 \), the derivative is \( 20x^3 \).
• For \( -6x^2 \), the derivative is \( -12x \).
• For the constant term \( 1 \), the derivative is \( 0 \).

Combining these, the second derivative is \( f''(x) = 20x^3 - 12x \). This gives us insight into how the slope (or the rate of change) of \( f(x) \) is evolving.

Power Rule

The power rule is a fundamental concept in calculus for finding derivatives. It simplifies the process of differentiation when dealing with polynomial functions. The power rule is expressed as follows: if you have a term \( x^n \), its derivative is \( nx^{n-1} \). This rule allows us to quickly find the derivatives of each term in a polynomial function without extensive computations.

For instance:

• If \( n = 5 \) in \( x^5 \), the derivative is \( 5x^4 \).
• If \( n = 3 \) in \( -2x^3 \), using the rule gives \( -6x^2 \).
• A constant term like \( x \) can be seen as \( x^1 \), and its derivative is \( 1 \).

The power rule is widely used due to its simplicity and effectiveness in dealing with polynomials, making it an essential tool for anyone studying calculus.