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Chapter 1: Problem 25

Find the slope of the curve \(y=x^{4}\) at \(x=2\).

Short Answer

Expert verified
The slope at x=2 is 32.

Step by step solution

01

Differentiate the function

To find the slope of a curve at a given point, start by differentiating the function with respect to x. The function given is \(y = x^4\). Differentiate it:\( \frac{dy}{dx} = \frac{d}{dx} (x^4) \).Using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \):\( \frac{dy}{dx} = 4x^3 \).
02

Substitute the given x-value into the derivative

Now that the derivative \( \frac{dy}{dx} = 4x^3 \) is found, substitute \(x = 2\) into the derivative to find the slope at that point. Replace \(x\) with \(2\):\( \frac{dy}{dx} \Bigg|_{x=2} = 4(2)^3 \).
03

Simplify the expression

Simplify the expression \(4(2)^3\):\( 4(2)^3 = 4 \times 8 = 32 \).Thus, the slope of the curve \( y = x^4 \) at \( x = 2 \) is 32.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differentiation
Differentiation is a fundamental concept in calculus that helps us find the rate at which a function is changing at any given point. Think of it as measuring how steep a curve is at a particular spot. This measure is called the derivative. To understand differentiation well, you should remember that it tells us the slope of the tangent line to the curve at a particular point.
  • The function we started with is \( y = x^4 \).
  • When finding the slope at a specific point, we differentiate the function with respect to \(x\).

So, differentiation is the tool we use to find out how a function is behaving moment to moment. That is, it tells us the instantaneous rate of change.
Power Rule
When differentiating functions of the form \( x^n \), where \( n \) is any real number, the power rule is a quick way to find the derivative. The power rule states: \( \frac{d}{dx}(x^n) = nx^{n-1} \). This is a very efficient rule that makes it easier to handle polynomials and other power functions.
  • Starting with \( y = x^4 \), we apply the power rule.
  • The derivative then becomes \( \frac{d}{dx}(x^4) = 4x^3 \).

The rule significantly simplifies the differentiation process. Anyone working on calculus problems should get comfortable with the power rule, as it is one of the most commonly used techniques. Remember, it works for any exponent, so you can use it whenever you deal with powers of \(x\).
Derivatives
Derivatives are the result of differentiation, telling us the rate of change of a function at any given point. They essentially give us the slope of the function at every point. In our example, after differentiating \( y = x^4 \) to get \( \frac{dy}{dx} = 4x^3 \), this derivative tells us how fast \( y \) is changing concerning \( x \).
  • To find the slope at \( x = 2 \), substitute \( x = 2 \) into the derivative.
  • So, \( 4(2)^3 = 4 \times 8 = 32 \).

The slope of the curve \( y = x^4 \) at \( x = 2 \) is 32. This means the tangent line to the curve at \( x = 2 \) rises steeply with a slope of 32. Understanding derivatives and how to compute them helps you analyze and interpret the behavior of different functions.

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Most popular questions from this chapter

Let \(P(x)\) be the profit (in dollars) from manufacturing and selling \(x\) cars. Interpret \(P(100)=90,000\) and \(P^{\prime}(100)=1200 .\) Estimate the profit from manufacturing and selling 99 cars.

Find the slope of the tangent line to the curve \(y=\left(x^{2}-15\right)^{6}\) at \(x=4 .\) Then write the equation of this tangent line.

Apply the three-step method to compute the derivative of the given function. $$f(x)=-x^{2}$$

After inspecting a sunken ship at a depth of 212 feet, a diver starts her slow ascent to the surface of the ocean, rising at the rate of 2 feet per second. Find \(y(t),\) the depth of the diver, measured in feet from the ocean's surface, as a function of time \(t\) (in seconds).

Use limits to compute the following derivatives. $$f^{\prime}(0), \text { where } f(x)=x^{3}+3 x+1$$
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