Problem 83 - Calculus for Biology and Medicine

Question

Find the first 10 derivatives of \(y=x^{5}\).

Step-by-Step Solution

Answer

The first 10 derivatives are: \(5x^4, 20x^3, 60x^2, 120x, 120, 0, 0, 0, 0, 0\).

Step 1: Understand the Derivative Formula

The derivative of a function, like a power function, applies the power rule \(\frac{d}{dx} x^n = nx^{n-1}\). We'll use this for the original function \(y = x^5\).

Step 2: Find the First Derivative

Using the power rule on \(y = x^5\), we differentiate to get \(y' = 5x^4\).

Step 3: Compute the Second Derivative

Apply the derivative again: \(y'' = \frac{d}{dx}(5x^4) = 20x^3\).

Step 4: Compute the Third Derivative

Differentiate again: \(y''' = \frac{d}{dx}(20x^3) = 60x^2\).

Step 5: Calculate the Fourth Derivative

Differentiate once more: \(y^{(4)} = \frac{d}{dx}(60x^2) = 120x\).

Step 6: Calculate the Fifth Derivative

Continue differentiating: \(y^{(5)} = \frac{d}{dx}(120x) = 120\).

Step 7: Calculate the Sixth Derivative

Differentiate again: \(y^{(6)} = \frac{d}{dx}(120) = 0\).

Step 8: Seventh to Tenth Derivative

Since the sixth derivative is zero, any further derivative of zero remains zero: \(y^{(7)} = 0\), \(y^{(8)} = 0\), \(y^{(9)} = 0\), \(y^{(10)} = 0\).

Key Concepts

Derivatives

In calculus, a derivative represents the rate at which a function is changing at any given point. Essentially, it's a tool to measure how a quantity changes for a small change in another quantity. For example, the derivative of a function concerning the variable 'x' helps understand how the function's value changes as 'x' changes.
The notation for the derivative of a function \(y\) with respect to \(x\) is \(\frac{dy}{dx}\) or simply \(y'\). For functions involving powers of \(x\), the process of differentiation allows us to apply rules, such as the power rule, to find this rate of change quickly. Derivatives are fundamental in many areas, including physics, engineering, and economics, as they provide insights into motion, optimizing processes, and understanding rates of change.

Power Rule

The power rule is a simple and essential formula used in calculus for differentiating power functions. It states that if you have a function \(x^n\), then the derivative of this function is \(nx^{n-1}\). This rule dramatically simplifies the process of finding derivatives especially for polynomial functions.
For example, when differentiating \(x^5\), its derivative becomes \(5x^4\). Notice that the exponent 5 of the original function moves in front as a coefficient, and we subtract one from the exponent to get the new power.
Using the power rule speeds up calculations and allows you to handle more complex expressions. It's a foundational technique every calculus student should master early on in their studies.

Higher-Order Derivatives

Higher-order derivatives refer to the derivatives of a derivative. The first derivative \(y'\) tells us the rate of change of the original function \(y\). The second derivative \(y''\) reflects how the first derivative is changing, providing information about the concavity and acceleration. Similarly, third, fourth, and higher derivatives can also be calculated.
For example, if you have \(y = x^5\) and you differentiate it to get \(y' = 5x^4\), you can continue differentiating to get \(y'' = 20x^3\), and so on, until you reach the tenth derivative, if required. In this exercise, the calculation ended with the sixth derivative because it was zero, and all subsequent derivatives of a constant zero function remain zero.
These higher derivatives are valuable for understanding the shape of graphs and the behavior of functions in more detail. They're used extensively in physics to describe motion and systems' response to forces, emphasizing their importance in applied contexts.