Problem 20 - Basic Engineering Mathematics

Question

If \(f(x)=4 x^{5}-2 x^{3}+x-3\), find \(f^{\prime \prime}(x)\)

Step-by-Step Solution

Answer

The second derivative is \(f''(x) = 80x^3 - 12x\).

Step 1: Find the First Derivative

To find the first derivative, denoted as \(f'(x)\), apply differentiation to each term in the function \(f(x) = 4x^5 - 2x^3 + x - 3\). For a term of the form \(ax^n\), the derivative is \(nax^{n-1}\). Thus, for each term:- The derivative of \(4x^5\) is \(5 \times 4x^{5-1} = 20x^4\).- The derivative of \(-2x^3\) is \(-3 \times 2x^{3-1} = -6x^2\).- The derivative of \(x\) is \(1x^{1-1} = 1\).- The derivative of a constant, \(-3\), is \(0\) as constants have no rate of change.

Step 2: Write the First Derivative

Combine the differentiated terms to get the first derivative:\[ f'(x) = 20x^4 - 6x^2 + 1 \]

Step 3: Find the Second Derivative

Differentiate the first derivative \(f'(x) = 20x^4 - 6x^2 + 1\) to find the second derivative, \(f''(x)\):- The derivative of \(20x^4\) is \(4 \times 20x^{4-1} = 80x^3\).- The derivative of \(-6x^2\) is \(2 \times -6x^{2-1} = -12x\).- The derivative of the constant \(1\) is \(0\).Thus, add these terms together to get \(f''(x)\).

Step 4: Write the Second Derivative

Combine the differentiated terms to find the second derivative:\[ f''(x) = 80x^3 - 12x \]

Key Concepts

First Derivative

The first derivative of a function helps us understand the rate of change, or how a function changes as its input changes. When you see a function like \( f(x) = 4x^5 - 2x^3 + x - 3 \), you are looking at a polynomial function comprised of multiple terms, each with its own x-power.
To find the first derivative, denoted \( f'(x) \), we perform differentiation. Differentiation is the process of finding the derivative. For polynomial terms, you use the rule that if \( ax^n \) is a term in your function, the derivative is \( nax^{n-1} \). This means you multiply the current power by the coefficient and reduce the power by 1.

• The term \( 4x^5 \) becomes \( 20x^4 \).
• The term \( -2x^3 \) becomes \( -6x^2 \).
• The term \( x \) simply becomes 1, because the x-power of 1 is reduced to 0.
• The constant term \(-3\) does not change, as constants have no variable x to differentiate. Its rate of change is 0.

Combining these results, the first derivative is \( f'(x) = 20x^4 - 6x^2 + 1 \). This represents the slope or steepness of the curve at any point on the function.

Differentiation

Differentiation is a key concept in calculus that involves finding the derivative of a function. The derivative tells us the rate at which a function's output value is changing at any given point. This is particularly useful in understanding how a polynomial function behaves as its input values change.
When differentiating a polynomial function, it’s important to remember some basic rules such as:

• Power Rule: To differentiate \( ax^n \), multiply \( a \) by \( n \) and reduce the power of \( x \) by one. For example, the derivative of \( 4x^5 \) is \( 5 \times 4x^{5-1} = 20x^4 \).
• Constants: The derivative of a constant term, such as \(-3\), is always 0 because it doesn’t change.

Applying these rules allows us to find the first derivative of any polynomial function. Differentiation helps us not only to find the first derivative but also further derivatives, like the second derivative, providing deeper insights into the function's behavior and acceleration.

Polynomial Function

A polynomial function is a mathematical expression consisting of terms that are powers of \( x \), each multiplied by a coefficient. Polynomial functions are versatile and can represent a wide range of curves and shapes, making them a central object of study in algebra and calculus.
In our exercise, we have the polynomial \( f(x) = 4x^5 - 2x^3 + x - 3 \). Each term is an instance of the general form \( ax^n \), where:

• \( 4x^5 \) is the leading term, with the highest power of 5.
• \( -2x^3 \) and \( x \) are middle terms with powers of 3 and 1, respectively.
• \(-3\) is the constant term that shifts the curve vertically.

Polynomial functions are known for their continuity and differentiability, which means their derivatives can be constructed simply by applying standard differentiation rules to each term. This makes it easier to explore properties like slopes, peaks, and concavities by finding successive derivatives.