Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-8\), find \(d^{3} y / d x^{3}\). $$ y=x^{5}+x^{4} $$

Step-by-Step Solution

Answer

The third derivative \(\frac{d^3y}{dx^3}\) is \(60x^2 + 24x\).

Step 1: Differentiate the First Time

To begin, let's find the first derivative of the function \(y = x^5 + x^4\). Apply the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Thus:\[ y' = \frac{dy}{dx} = 5x^4 + 4x^3 \]

Step 2: Differentiate the Second Time

Now, differentiate the first derivative to get the second derivative. Again, apply the power rule:\[ y'' = \frac{d^2y}{dx^2} = 20x^3 + 12x^2 \]

Step 3: Differentiate the Third Time

Finally, differentiate the second derivative to obtain the third derivative. Use the power rule once more:\[ y''' = \frac{d^3y}{dx^3} = 60x^2 + 24x \]

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus for differentiation. It simplifies the process of finding derivatives for functions of the form \( x^n \). In essence, the power rule states: "Bring down the exponent as a coefficient and subtract one from the exponent."

This can be represented mathematically as:

• If \( y = x^n \), then the derivative \( \frac{dy}{dx} = nx^{n-1} \).

For example, applying the power rule to differentiate \( x^5 \) gives \( 5x^4 \), and differentiating \( x^4 \) results in \( 4x^3 \).

By applying this rule successively to the polynomial \( y = x^5 + x^4 \), you can quickly find the derivatives needed. It's particularly useful for polynomials because each term can be differentiated independently.

Differentiation

Differentiation is the process in calculus used to find the rate at which a function is changing. This is especially important in understanding how variables are related by a function.

To differentiate a function once is to find its first derivative. This derivative represents the instantaneous rate of change of the function with respect to one of its variables. For the function \( y = x^5 + x^4 \), when you perform differentiation:

• First derivative \( y' = 5x^4 + 4x^3 \).
• Second derivative \( y'' = 20x^3 + 12x^2 \).
• Third derivative \( y''' = 60x^2 + 24x \).

Each differentiation step involves applying the power rule to each term in the polynomial. This methodically reveals how the polynomial's slope, concavity, and rate of change evolve.

Calculus Problem Solving

Calculus problem-solving often involves systematically applying rules and methods to find derivatives, like the example problem, where you need to find the third derivative. Breaking down the problem into steps is key:

• First, find the first derivative using the power rule.
• Then, find the second derivative by differentiating the first derivative.
• Finally, determine the third derivative by differentiating the second derivative.

This structured approach helps in minimizing errors and understanding the process better. Each derivative can provide insight into the behavior of a function, such as slope and curvature, which is why breaking it down into manageable steps is crucial for clarity.

Approaching calculus systematically not only improves problem-solving skills but also deepens comprehension of mathematical relationships and the potential applications of derivatives.