Problem 54 - Calculus and its applications

Question

Find \(y^{\prime}\) $$ \text { If } y=x^{3}+2 x-5, \text { find }\left.\frac{d y}{d x}\right|_{x=-2} $$

Step-by-Step Solution

Answer

The derivative \(\frac{dy}{dx}\) at \(x = -2\) is 14.

Step 1: Differentiate the Function

First, find the derivative of the given function \(y = x^3 + 2x - 5\) with respect to \(x\). Use the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\), and the derivative of a constant is 0. Thus, the derivative of \(y\) is \(\frac{dy}{dx} = 3x^2 + 2\).

Step 2: Substitute \(x = -2\)

Now that we have the general form of the derivative \(\frac{dy}{dx} = 3x^2 + 2\), substitute \(x = -2\) into this expression to find the derivative at that specific point. Here we calculate: \(\frac{dy}{dx} = 3(-2)^2 + 2\).

Step 3: Simplify the Expression

Work out the squared term and multiply to simplify the expression. \((-2)^2 = 4\), so the derivative is \(3 \times 4 + 2 = 12 + 2 = 14\).

Key Concepts

Differentiation

Differentiation is the process used in calculus to find the rate at which a function is changing at any given point. In simpler terms, it helps us understand how a function behaves locally, whether it’s increasing, decreasing, or remaining constant. This is done by finding the derivative of the function.
For the function given in the exercise, the goal is to differentiate it with respect to the variable \(x\). When you differentiate, you essentially use different rules, such as the power rule, to find an expression that describes the slope of the tangent line at any point on the graph of the function. This slope is what we refer to as the derivative.

Power Rule

The power rule is a fundamental tool in the process of differentiation. It greatly simplifies finding the derivatives of polynomial functions. The power rule is stated as follows: the derivative of \(x^n\) is \(nx^{n-1}\), where \(n\) is a constant power.
When applying the power rule, remember:

• The exponent of \(x\) becomes multiplied by the coefficient.
• The new exponent becomes \(n - 1\).

This rule also applies to each term within a polynomial separately. So, for the function \(y = x^3 + 2x - 5\), use the power rule for each term:

• The derivative of \(x^3\) is \(3x^2\).
• The derivative of \(2x\) is \(2\), since it's equivalent to \(2x^1\).
• The derivative of a constant like \(-5\) is \(0\).

So, the derivative of the function is \(3x^2 + 2\). By mastering the power rule, you can easily tackle polynomial derivatives efficiently.

Finding Derivatives at a Point

Once you have the derivative of a function, you can find the derivative at a specific point, essentially answering the question, "What is the slope of the tangent line to the curve at this point?"
In this exercise, we've found the derivative of \(y = x^3 + 2x - 5\), which is \(3x^2 + 2\). To find the derivative at \(x = -2\), we substitute \(x = -2\) into the derivative:

• Calculate \(3(-2)^2 + 2\).
• Simplify \((-2)^2\) to \(4\), so \(3 \times 4\) becomes \(12\).
• Add \(2\) to get \(14\).

Therefore, the derivative at \(x = -2\) is \(14\). This means that at \(x = -2\), the tangent to the curve has a slope of \(14\), indicating how steeply the graph is rising at that particular point.