Problem 21 - Calculus Early Transcendentals

Question

Find \(d y /\left.d x\right|_{x=1}\) $$ y=1+x+x^{2}+x^{3}+x^{4}+x^{5} $$

Step-by-Step Solution

Answer

15

Step 1: Understand the Problem

We need to find the derivative of the function \(y = 1 + x + x^2 + x^3 + x^4 + x^5\) with respect to \(x\), and then evaluate this derivative at \(x = 1\). This is often referred to as finding \(\frac{dy}{dx}\bigg|_{x=1}\).

Step 2: Differentiate the Function

To find the derivative of \(y\) with respect to \(x\), we differentiate each term of the function separately:- The derivative of \(1\) is \(0\).- The derivative of \(x\) is \(1\).- The derivative of \(x^2\) is \(2x\).- The derivative of \(x^3\) is \(3x^2\).- The derivative of \(x^4\) is \(4x^3\).- The derivative of \(x^5\) is \(5x^4\).Thus, \(\frac{dy}{dx} = 0 + 1 + 2x + 3x^2 + 4x^3 + 5x^4\), which simplifies to \(1 + 2x + 3x^2 + 4x^3 + 5x^4\).

Step 3: Evaluate the Derivative at \(x=1\)

Now, we substitute \(x = 1\) into the derivative \(1 + 2x + 3x^2 + 4x^3 + 5x^4\) and simplify:\(1 + 2(1) + 3(1)^2 + 4(1)^3 + 5(1)^4 = 1 + 2 + 3 + 4 + 5\).

Step 4: Simplify the Expression

Add all the terms from the previous step: \(1 + 2 + 3 + 4 + 5 = 15\).So, \(\frac{dy}{dx}\bigg|_{x=1} = 15\).

Key Concepts

Polynomial Differentiation

Polynomial differentiation is a technique used to find the derivative of a polynomial function. Almost every function in algebra that you will encounter can be expressed as a polynomial sum, like the function in our original exercise: \[ y = 1 + x + x^2 + x^3 + x^4 + x^5 \]In polynomial differentiation, we simply apply the derivative rules to each term separately, making it a very straightforward process. Each term in the polynomial has its own power and coefficient. By using the power rule, which we will discuss further, the derivative of each term is easy to calculate. This rule states: - The derivative of a term of the form \(x^n\) where \(n\) is a constant, is \(nx^{n-1}\). Let's look at our polynomial:- For the constant \(1\), the derivative is \(0\) because constant terms have no \(x\) term.- For \(x\), the derivative is \(1\) because it's the same as \(x^1\).- For \(x^2\), the derivative is \(2x\).- For \(x^3\), the derivative is \(3x^2\). - For \(x^4\), the derivative is \(4x^3\).- For \(x^5\), the derivative is \(5x^4\).Using these steps, we can form the derivative \(1 + 2x + 3x^2 + 4x^3 + 5x^4\) for the entire polynomial.

Evaluation of Derivative

Once you have computed the derivative of a polynomial, often you need to evaluate it at a specific point. This means substituting the value of \(x\) into the derivative function.For our exercise, we're tasked with evaluating the derivative at \(x = 1\). This involves substituting \(1\) for \(x\) in our derived polynomial:\[ 1 + 2x + 3x^2 + 4x^3 + 5x^4\]Substitute \(x = 1\):- The first term becomes: \(1\)- The second term becomes: \(2(1) = 2\)- The third term becomes: \(3(1)^2 = 3\)- The fourth term becomes: \(4(1)^3 = 4\)- The fifth term becomes: \(5(1)^4 = 5\)Add these values together: \[ 1 + 2 + 3 + 4 + 5 = 15 \]The result of this evaluation, at \(x = 1\), tells us the slope of the tangent line to the curve at that point. Thus, in our exercise, the derivative evaluated at \(x = 1\) is \(15\).

Basic Differentiation Rules

Differentiation is a fundamental operation in calculus, and understanding the basic rules simplifies the process considerably. The two main rules we applied in our exercise are the power rule and the sum rule.

• **Power Rule:** This is used to find the derivative of a term with an exponent. For \(x^n\), the derivative is \(nx^{n-1}\). You reduce the power by one and multiply by the original power number.
• **Constant Rule:** The derivative of any constant is \(0\), as constants have no rate of change.
• **Sum Rule:** Differentiation operates on sums of functions by differentiating each term independently. This means that for a sum like \(u + v\), the derivative is \(du/dx + dv/dx\).

These rules form the building blocks of using differentiation on polynomial functions. When you break down these operations: - Each term in a polynomial is treated as an independent function.- Differentiation rules are applied to each term individually.- The results are then summed.This allows you to simplify complex polynomials into easily calculable derivatives, like the one seen in our problem: \(1 + 2x + 3x^2 + 4x^3 + 5x^4\). By mastering these fundamental rules, differentiation becomes an accessible and efficient process.