Problem 10 - Thomas Calculus

Question

Find the first and second derivatives. $$ y=4-2 x-x^{-3} $$

Step-by-Step Solution

Answer

First derivative: \(y' = -2 + 3x^{-4}\); Second derivative: \(y'' = -12x^{-5}\).

Step 1: Identify the Function

The function given is \[ y = 4 - 2x - x^{-3} \] This function includes a constant term, a linear term, and a term with a negative exponent. Our task is to find the first and second derivatives of this function.

Step 2: Apply the Derivative Rules

To find the first derivative, apply the power rule for derivatives. The derivative of a constant is 0, the derivative of \(-2x\) is \(-2\), and the derivative of \(x^{-3}\) is \(-3x^{-4}\) (using the rule that if \(y=x^n\), then \(y'=nx^{n-1}\)).

Step 3: Calculate the First Derivative

Differentiate each term of the function separately: - The derivative of \(4\) is \(0\).- The derivative of \(-2x\) is \(-2\).- The derivative of \(-x^{-3}\) is \(3x^{-4}\).Combining these, the first derivative \(y'\) is:\[ y' = 0 - 2 + 3x^{-4} \]\[ y' = -2 + 3x^{-4} \]

Step 4: Apply the Derivative Again

Next, apply the derivative rules to the first derivative to find the second derivative. Recall that the derivative of \(x^n\) is \(nx^{n-1}\). Calculate derivatives for each term in the first derivative.

Step 5: Calculate the Second Derivative

Differentiate the first derivative:- The derivative of \(-2\) is \(0\) (since it's a constant).- The derivative of \(3x^{-4}\) is \(-12x^{-5}\) (using the power rule \(3 \cdot (-4)x^{-4-1}\)).Thus, the second derivative \(y''\) is:\[ y'' = 0 - 12x^{-5} \]\[ y'' = -12x^{-5} \]

Step 6: State the Derivatives

The derivatives of the function \(y = 4 - 2x - x^{-3}\) are:- First derivative: \(y' = -2 + 3x^{-4}\).- Second derivative: \(y'' = -12x^{-5}\).

Key Concepts

First Derivative

The first derivative of a function tells us how the function's output changes as the input changes. This is also known as the slope of the tangent line at any point on the curve of the function. To find the first derivative of a given function, we use several rules of differentiation. Among them, the power rule is highly important. In our example, the function is given as \( y = 4 - 2x - x^{-3} \).

• The constant term \(4\) has a derivative of \(0\) because constants do not change, and thus their rate of change is zero.
• The term \(-2x\) is linear. Its derivative is \(-2\). This is because the derivative of \(x\) is \(1\), and multiplying it by \(-2\) gives us \(-2\).
• The term \(-x^{-3}\) follows the power rule. This rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Applying this, the derivative of \(-x^{-3}\) becomes \(3x^{-4}\).

Bringing all these terms together, the first derivative \(y'\) is \[ y' = -2 + 3x^{-4} \]. This equation helps us understand how each part of the function contributes to its rate of change.

Second Derivative

The second derivative provides insight into the curvature or concavity of the original function. It lets us know if the function is curving upwards or downwards. Understanding the concavity can tell us more about the behavior of the graph. For instance, it helps identify inflection points where the function changes from being concave up to concave down, or vice-versa.From our previous step, we have the first derivative: \( y' = -2 + 3x^{-4} \).

• The second derivative is found by differentiating the first derivative. The derivative of the constant \(-2\) is \(0\), again because the rate of change of a constant is zero.
• For the term \(3x^{-4}\), using the power rule, the derivative becomes \(-12x^{-5}\). This rule applies as before: \(3 \cdot (-4)x^{-4-1}\).

Thus, the second derivative \(y''\) is \[ y'' = -12x^{-5} \]. This tells us that the original function's graph is influenced by the rates at which these terms change, particularly impacting its shape and direction of curvature.

Power Rule

The power rule is a fundamental tool in calculus for finding derivatives easily, especially for polynomial functions. It simplifies the differentiation process significantly by providing a quick way to determine how each term in the function changes.When applying the power rule, remember:

• If you have a term in the form \(x^n\), the derivative is \(nx^{n-1}\).
• This rule applies to both positive and negative exponents, making it versatile for various types of functions.
• It's critical to multiply by the current exponent and then decrease the power by one.

For instance, in the function \( y = 4 - 2x - x^{-3} \), the terms are treated individually:- The derivative of \(x^1\) becomes 1, then multiplied by the coefficient, yielding \(-2\).- The derivative of \(x^{-3}\) becomes \(-3x^{-4}\) by applying the power rule with a negative exponent.This rule is especially valuable in calculus due to its efficiency and simplicity in breaking down complex polynomial expressions into manageable parts for differentiation.