Question

Find the slope of the curve \(y=x^{5}\) in the point whose abscissa is 2 .

Short answer

Answer: The slope of the curve at the given point is 80.

Step-by-step solution

Find the Derivative of the Function

To find the derivative of the function, we will use the power rule, which states that the derivative of a function \(f(x) = x^{n}\), where n is a constant, is given by \(f'(x) = nx^{n-1}\). We have the function \(y = x^{5}\), so we can find the derivative as: $$y'(x) = 5x^{5-1} = 5x^{4}$$.

Evaluate the Derivative at the Given Point

Now that we have found the derivative of the function, we can evaluate it at the point where the abscissa is 2. This will give us the slope of the curve at that point: $$y'(2) = 5 (2^{4}) = 5 (16) = 80$$.

Determine the Slope of the Curve

As a result, the slope of the curve \(y = x^{5}\) at the point where the abscissa is 2 is found by evaluating the derivative, which in this case equals to 80.

Key concepts

Derivative of a Function

When we talk about the derivative of a function, we're discussing a fundamental concept in calculus that measures how a function changes as its input changes. It's like capturing a snapshot of a moving object's speed at a specific moment in time.

The derivative tells us about rates of change. For instance, if you're driving and your speedometer reads 60 miles per hour, this is a rate of change: your position is changing at a rate of 60 miles each hour. In calculus, if we have a function that represents your position over time, the derivative of that function at any given point tells us your speed at that moment.

In mathematical terms, the derivative of a function at a certain point is the limit of the average rate of change of the function in an infinitely small interval around that point. This limit, if it exists, gives us instant rate of change of the function with respect to its variable, which can be visualized as the slope of the tangent line to the function's graph at that point. To find this value, we usually have to apply various rules and techniques of differentiation, which brings us to our next concept.

Power Rule for Derivatives

The power rule for derivatives is one of the most straightforward and frequently used rules for differentiation. It's a quick method to find the derivative when you're dealing with functions that are powers of a variable, like f(x) = xⁿ, where n is any real number.

The power rule states this: To find the derivative of xⁿ, multiply n by x to the power of n-1. So, the derivative—also denoted as f'(x) or y'—is nx^n-1.

For example, if you have a function like f(x) = x⁵, using the power rule, its derivative f'(x) would be 5x⁴. It's also important to remember that this rule only applies when x is raised to a constant power. When you encounter such functions, remembering the power rule makes differentiation a much smoother process.

Evaluating Derivatives

Once we have the derivative of a function, the next step is often to evaluate the derivative at a specific point. This process involves substituting a particular value for the variable in the derivative formula to calculate the slope of the tangent line at that point on the original function's graph.

Evaluating the derivative at a particular point gives us valuable insights: It could be the instantaneous rate of change in a physical problem, the velocity of an object at a certain time, or, in the case of our exercise, the slope of the curve at a specific point.

Applying the Concept to an Exercise

In our textbook exercise, after finding the derivative of y = x⁵ to be 5x⁴, we evaluated it when x = 2. This meant substituting 2 into the derivative to get 5*(2⁴), which simplifies to 80. Hence, we determined that the slope of the curve at the point where the abscissa is 2 is 80.

This ability to evaluate derivatives at specific points helps us understand the behavior of functions at those points, which is crucial for solving a wide array of problems in mathematics, physics, engineering, and economics.