Question

In Exercises,Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } y=(x+1)(x+2)(x+3)(x+4), \text { then } \frac{d^{5} y}{d x^{5}}=0 $$

Short answer

The statement is true as the fifth derivative of function \(y=(x+1)(x+2)(x+3)(x+4)\) is indeed 0.

Step-by-step solution

Find initial equation

Take the given function, \(y=(x+1)(x+2)(x+3)(x+4)\). This is a polynomial of degree 4.

Find the first derivative

The first derivative of the function \(y\) is given by \(y' = 4x^3 + 30x^2 + 76x + 64\)

Find the second derivative

The second derivative of the function \(y\) is given by \(y'' = 12x^2 + 60x + 76\)

Find the third derivative

The third derivative of the function \(y\) is \(y''' = 24x + 60\)

Find the fourth derivative

The fourth derivative of the function \(y\) is given by \(y'''' = 24\)

Find the fifth derivative

The fifth derivative of the function \(y\) is \(y''''' = 0\) since the derivative of a constant is 0.

Conclude the proof

Since the fifth derivative of the given function is indeed 0, the statement is proven to be true.

Key concepts

Higher-Order Derivatives

When we study calculus and specifically differentiation, we often begin with finding the first derivative of a function, which provides us with the slope of the tangent line at any point on the function's curve. However, the exploration does not stop there.

Higher-order derivatives refer to derivatives taken multiple times. For instance, after finding the first derivative, which is also known as the velocity in physics, we can also find the second derivative. The second derivative tells us about the acceleration, or the rate at which the velocity is changing. The process can continue, leading us to third, fourth, and so on, known as the third-order derivative, fourth-order derivative, respectively.

In the exercise provided, we see a polynomial function, and we are tasked with finding up to the fifth derivative, labeled as \( \frac{d^{5} y}{d x^{5}} \). With each successive differentiation, we get values that represent the changing rate of the previous derivative's graph. The moment we reach a constant or a zero while differentiating, any higher-order derivatives will also be zero, since differentiating a constant always yields zero.

Polynomial Functions

Polynomial functions are algebraic expressions that include terms in the form of \( a_nx^n \) where \( n \) is a non-negative integer and \( a_n \) is a coefficient. They are one of the most extensively studied objects in algebra and calculus due to their simplicity and the rich variety of patterns they can model.

For example, in our exercise, \( y=(x+1)(x+2)(x+3)(x+4) \) is a polynomial function of degree four, since the highest power of \( x \) after expanding the product is \( x^4 \). This function is smooth and continuous everywhere, and it can be differentiated any number of times. However, after a certain number of differentiations, which corresponds to the degree of the polynomial, the derivative becomes zero, as evidenced in the problem solution.

Calculus

Calculus, a significant branch of mathematics, is all about change. It has two major areas: differential calculus (concerned with the concept of a derivative) and integral calculus (concerned with the concept of an integral).

Through differential calculus, we analyze instant rates of change, slopes of curves, and the behavior of graphs. As seen in the original exercise, we use differential calculus to find the rate of change of a quantity - in this case, the function \( y \) as it relates to \( x \). Each subsequent derivation gives us more insight into the behavior of the function's graph. The process of finding these derivatives is known as differentiation, a fundamental tool used to solve a wide array of problems in both mathematics and physics.

Derivative of a Constant

A fundamental rule in differentiation is that the derivative of a constant is always zero. This is because a constant does not change; therefore, it has no rate of change, which is what a derivative measures. No matter how many times you differentiate a constant value, the result will always be zero.

In the context of our original exercise, once we reach the fourth derivative of the function \( y \), we obtain \( y'''' = 24 \) – a constant. So, when we take the fifth derivative, \( y''''' \), we get zero, affirming the statement given in the problem. This concept is vital to understand because it informs us that polynomial functions of a certain degree will eventually result in a constant derivative, after which all higher-order derivatives will yield zero.