Problem 11 - The Calculus with Analytic Geometry

Question

Find \(D_{x}^{3} y\) if \(y=x^{4}-2 x^{2}+x-5\).

Step-by-Step Solution

Answer

The third derivative is \( y''' = 24x \).

Step 1 - Identify the Function

Start with the given function: \[ y = x^4 - 2x^2 + x - 5 \]

Step 2 - Find the First Derivative

Find the first derivative of the function with respect to x. Differentiate each term separately: \[ y' = D_x (x^4) - D_x (2x^2) + D_x (x) - D_x (5) \] Using the power rule \( D_x (x^n) = nx^{n-1} \): \[ y' = 4x^3 - 4x + 1 \]

Step 3 - Find the Second Derivative

Now, differentiate the first derivative to find the second derivative: \[ y'' = D_x (4x^3) - D_x (4x) + D_x (1) \] Using the power rule again: \[ y'' = 12x^2 - 4 \]

Step 4 - Find the Third Derivative

Differentiate the second derivative to find the third derivative: \[ y''' = D_x (12x^2) - D_x (4) \] Using the power rule one more time: \[ y''' = 24x \]

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the rate at which a function is changing at any given point. This rate of change is called the derivative. Differentiation allows us to determine the slope of a curve at any point, which is extremely useful for understanding how functions behave.
To differentiate a function:

• Identify the function you need to differentiate.
• Apply differentiation rules, such as the power rule, product rule, quotient rule, or chain rule.
• Simplify the resulting expression if needed.

Let's take the function in the exercise: y=x^{4}-2x^{2}+x-5. By differentiating each term individually, we get new functions representing the rates of change.

Power Rule

The power rule is a basic yet essential tool in differentiation and helps simplify the process. The power rule states: \( D_x (x^n) = nx^{n-1} \).
Let’s break it down:

• ‘n’ is the exponent of the term.
• To find the derivative, multiply the term by its exponent, ‘n’.
• Subtract one from the exponent.

For example, if we have a term like \( x^4 \), its derivative will be \(4x^{3} \). This rule is repeatedly used in our original exercise for differentiating higher-order derivatives.
Applying the power rule step-by-step, we derived: \( y' = 4x^3 - 4x + 1 \) for the first derivative,\( y'' = 12x^2 - 4 \) for the second derivative, and finally, \( y''' = 24x \) for the third derivative.

Higher-Order Derivatives

Higher-order derivatives are derivatives of derivatives. The first derivative, usually denoted as \( y' \) or \( f'(x) \), reflects the rate of change of the original function.
The second derivative \( y'' \) shows the rate of change of the first derivative. Higher-order derivatives provide deeper insights into the function's behavior.
• The third derivative, noted as \( y''' \) or \( f'''(x) \), helps us understand the pattern of curvature and acceleration. Higher-order derivatives can continue indefinitely, offering increasingly detailed views of how a function changes.
In our exercise, we found the third derivative of \( y=x^{4}-2x^{2}+x-5 \) by differentiating step-by-step:

• The first derivative: \( y' = 4x^3 - 4x + 1 \)
• The second derivative: \( y'' = 12x^2 - 4 \)
• The third derivative: \( y''' = 24x \)

Understanding each derivative layer by layer helps solidify grasping complex functions.