Question

Derivative at a Given Point. Find the slope of the tangent to the function \(y=x-x^{2}\) at \(x=2\).

Short answer

The slope of the tangent to the function at \(x=2\) is \(-3\).

Step-by-step solution

Identify the Function

The given function is defined as \(y = x - x^2\). The goal is to find the derivative of the function, which will give us the slope of the tangent at any given value of \(x\).

Differentiate the Function

To find the derivative of the function \(y = x - x^2\), we will use the power rule for differentiation. The power rule states that the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\). Thus, the derivative of \(x\) is 1 and the derivative of \(-x^2\) is \(-2x\). The combined derivative of the function is \(y' = 1 - 2x\).

Evaluate the Derivative at the Given Point

The slope of the tangent at the point where \(x=2\) is found by evaluating the derivative at that point. Substituting \(x=2\) into the derivative, we get \(y' = 1 - 2(2) = 1 - 4 = -3\).

Key concepts

Slope of Tangent

Understanding the slope of a tangent to a curve at a given point is paramount in the study of calculus. The slope represents the steepness or incline of the tangent line, which is a line that just grazes the curve at that single point without cutting through it. When you calculate the slope of this tangent, you are essentially finding the instantaneous rate of change of the function at that point.

To find the slope of the tangent, we look for the derivative of the function. For the given function, \( y = x - x^2 \), the tangent line at \( x=2 \) will have a slope that equals the derivative of the function evaluated at \( x=2 \). This process is conveyed in the solution steps and reveals that the slope, in this case, is \( -3 \). This means that at \( x=2 \) the curve is decreasing; for every unit increase in \( x \), \( y \) decreases by 3. It's like descending a hill that drops 3 feet vertically for every foot traveled horizontally.

Power Rule for Differentiation

The power rule for differentiation is a cornerstone in computing derivatives. It simplifies the process for handling polynomial functions, like the one in our exercise, \( y = x - x^2 \). According to the power rule, to differentiate \( x^n \), where \( n \) is any real number, you multiply \( x \) by \( n \) and decrease the exponent by one, resulting in \( nx^{n-1} \).

Applying this rule, the derivative of \( x \) is \( 1x^{1-1} \), which simplifies to \( 1 \), as any number to the zeroth power equals 1. The derivative of \( -x^2 \) becomes \( -2x^{2-1} \) or \( -2x \). Combining these, the derivative of the entire function is \( 1 - 2x \), which allows us to find the slope of the tangent at any point along \( y \) by substituting the corresponding \( x \) value.

Evaluating Derivatives

Evaluating derivatives at a particular point is a practical application of derivatives, affirming their importance in calculus. Once we have the derivative of a function, we can find the rate of change at any specific point by substituting the \( x \) value of that point into the derivative.

In the case of the function \( y = x - x^2 \), after applying the power rule, we determined the derivative to be \( y' = 1 - 2x \). To evaluate this derivative at \( x=2 \)—the given point—we replace \( x \) with 2, which gives us \( y' = 1 - 2(2) \) or \( y' = 1 - 4 \) which simplifies to \( -3 \). This result tells us the slope of the tangent, but also serves as a basis for other applications, like determining maximum or minimum values, or constructing a tangent line equation. Thus, mastering this fundamental concept of evaluating derivatives paves the way for deeper understanding and problem-solving in calculus.