Question

Find \(D_{x} y\) using the rules of this section. $$ y=x^{12}+5 x^{-2}-\pi x^{-10} $$

Short answer

\(D_{x} y = 12x^{11} - 10x^{-3} + 10\pi x^{-11}\).

Step-by-step solution

Differentiate the first term

The first term of the function is \(x^{12}\). To differentiate this with respect to \(x\), we use the power rule: if \(y = x^n\), then \(\frac{dy}{dx} = nx^{n-1}\). Applying this rule, we get \(\frac{d}{dx}[x^{12}] = 12x^{11}\).

Differentiate the second term

The second term of the function is \(5x^{-2}\). Again, applying the power rule, we get \(\frac{d}{dx}[5x^{-2}] = 5 \cdot (-2)x^{-3} = -10x^{-3}\).

Differentiate the third term

The third term is \(-\pi x^{-10}\). Using the power rule here, along with the constant multiplication rule, we differentiate: \(\frac{d}{dx}[-\pi x^{-10}] = -\pi \cdot (-10) x^{-11} = 10\pi x^{-11}\).

Combine the derivatives

To find \(D_{x} y\), we combine the derivatives from each term. Thus, \(D_{x} y = 12x^{11} - 10x^{-3} + 10\pi x^{-11}\).

Key concepts

Power Rule

The Power Rule is a cornerstone in differentiation, which simplifies finding the derivative of functions that are powers of a variable. If you have a term like \(x^n\), where \(n\) is any real number, you can determine its derivative using the rule: \(\frac{d}{dx}[x^n] = nx^{n-1}\). This means you multiply by the power \(n\) and then reduce the power by one.

For instance, in the term \(x^{12}\), applying the power rule gives \(12x^{11}\). This rule is incredibly useful because it streamlines the differentiation process, avoiding the need for complex algebra. It's a direct, mechanical step that ensures efficiency when calculating derivatives.

• Use: To differentiate polynomial terms.
• Benefit: Simplifies calculation, especially when the power is a whole number.

Constant Multiplication Rule

The Constant Multiplication Rule explains how to differentiate terms that are multiplied by constants. When you have a constant \(c\) multiplied by a function \(f(x)\), the derivative is simply the constant multiplied by the derivative of the function: \(\frac{d}{dx}[cf(x)] = c \frac{d}{dx}[f(x)]\). This makes it easy to handle constants in differentiation.

For example, when differentiating \(5x^{-2}\), the constant \(5\) is simply multiplied by the derivative of \(x^{-2}\) to give \(-10x^{-3}\). This rule simplifies the differentiation process and helps in combining with other rules like the Power Rule.

• Key Point: The constant remains and multiplies the derivative.
• Application: Essential when terms include coefficients.

Calculus

Calculus is the branch of mathematics focused on continuous change. It's divided into two main components: differential calculus, which deals with the rates of change (slopes and velocities), and integral calculus, which concerns accumulation (areas and volumes).

Differential calculus, as seen in our example, helps find how functions behave, grow, and shrink by examining their rates of change. When you differentiate a function, you're finding its derivative, the primary tool in calculus for understanding these dynamic changes.

• Purpose: To offer a mathematical framework for analyzing motion and change.
• Components: Derivatives (for change) and Integrals (for accumulation).

Derivative Calculation

Finding a derivative is essentially calculating how a function changes as its input changes. In our exercise, we applied the Power Rule and Constant Multiplication Rule to different terms of the function to determine how each part changes.

Steps to derivative calculation include:

• Identify terms of the function.
• Apply differentiation rules like Power Rule and Constant Multiplication.
• Simplify your results as needed and combine.

The result, \(D_x y = 12x^{11} - 10x^{-3} + 10\pi x^{-11}\), provides a complete picture of the function's rate of change. Calculating derivatives is a core skill in calculus, providing insights into function behavior, optimization, and modeling real-world situations.