Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\) using the rules of this section. $$ y=\pi x^{7}-2 x^{5}-5 x^{-2} $$

Step-by-Step Solution

Answer

\( D_x y = 7\pi x^{6} - 10x^{4} + 10x^{-3} \)

Step 1: Differentiate Each Term

The function given is \( y = \pi x^7 - 2x^5 - 5x^{-2} \). We use the power rule to differentiate each term separately. Recall that the power rule states: \( \frac{d}{dx} [x^n] = nx^{n-1} \).Differentiate each term as follows:For \( \pi x^7 \), the derivative is \( 7\pi x^{6} \).For \( -2x^5 \), the derivative is \( -10x^{4} \).For \( -5x^{-2} \), the derivative is \( 10x^{-3} \).

Step 2: Combine the Derivatives

Now, combine the derivatives of each term to find \( D_x y \). Sum up the separate derivatives from Step 1:\[ D_x y = 7\pi x^{6} - 10x^{4} + 10x^{-3} \].

Key Concepts

Power Rule

The power rule is a fundamental tool in differentiation, particularly with polynomial functions. It's a quick and efficient way to find the derivative of a term where the variable is raised to a power. Here’s how it works: if you have a term in the form of \(x^n\), its derivative is found by multiplying the power \(n\) by the coefficient of \(x\) and then reducing the power by one. So, the derivative of \(x^n\) is \(nx^{n-1}\). This makes it incredibly easy to differentiate terms like \(3x^2\) or \(5x^4\).

The power rule is also applicable to terms with constant coefficients, like \( \pi x^7 \). The coefficient remains as part of the derivative, making the derivative of \( \pi x^7 \) equal to \(7\pi x^6\). Notice how the power has decreased from 7 to 6, while the coefficient \( \pi \) remains unchanged.

Derivative of Polynomial Functions

Polynomial functions are algebraic expressions made up of variables and coefficients, involving only the operations of addition, subtraction, and multiplication, and with non-negative integer exponents. Differentiating polynomial functions means applying the power rule to each term separately. This is because derivatives are linear operators, which simplify the process to dealing term-by-term.

Consider a function like \( y = x^3 + 2x^2 - x + 5 \). To find its derivative, apply the power rule to each of its terms:

• The derivative of \(x^3\) is \(3x^2\).
• The derivative of \(2x^2\) is \(4x\).
• The derivative of \(-x\) is \(-1\).
• Constant terms like '5' become 0 in differentiation.

Combine these to get the derivative \(y' = 3x^2 + 4x - 1\). Differentiating each term and summing them gives the overall derivative of the polynomial function.

Basic Differentiation Techniques

Basic differentiation techniques revolve around applying simple rules like the power rule, and then summing up the derivatives of individual terms. Once you understand the mechanics, it's about practice and application.

Here are some essentials to keep in mind:

• Differentiation is a linear operation. This means you can differentiate terms separately and then sum them up.
• Constant multiples of functions remain through differentiation. For example, \(3x^2\) differentiated is \(6x\), where 3 is a constant multiple.
• Derivatives of constants are always zero. If you have a term like '+7', its derivative is '0'.

Practicing with simple polynomial functions can reinforce these concepts, making you adept at recognizing patterns and efficiently calculating derivatives. Remember, consistency is key; apply the rules correctly every step of the way, and double-check your work by substituting back to verify the derivatives match the function's behavior.