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Chapter 7: Problem 59

Find the derivative of \(y\) with respect to the given independent variable. $$y=x^{n}$$

Short Answer

Expert verified
The derivative of \( y = x^n \) is \( nx^{n-1} \).

Step by step solution

01

Understanding the Problem

We need to find the derivative of the function \( y = x^n \) with respect to \( x \). The exponent \( n \) is a constant.
02

Identifying the Rule to Apply

For finding derivatives of functions of the form \( x^n \), we use the power rule. The power rule states that if \( y = x^n \), then the derivative is \( \frac{dy}{dx} = nx^{n-1} \).
03

Applying the Power Rule

Using the power rule, we differentiate: \( \frac{dy}{dx} = n \cdot x^{n-1} \).
04

Simplifying the Derivative

Since \( n \) is a constant, computation is direct. Therefore, the derivative is \( n \cdot x^{n-1} \), which is already in its simplest form.

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Key Concepts

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

Power Rule in Calculus
The power rule is one of the most essential tools in calculus, especially when dealing with differentiation. It gives a straightforward method to determine the derivative of functions in the form of a power expression, like \( y = x^n \).
  • The rule applies when you have a variable raised to a constant exponent.
  • To differentiate \( x^n \), you bring the exponent "\( n \)" down as a coefficient.
  • Then, you subtract one from the exponent itself.

Mathematically, this is shown as:
\[ \frac{d}{dx}x^n = n \cdot x^{n-1} \]
This simplification makes it easy and quick to work with polynomial functions and is the reason why the power rule is often among the first differentiation rules taught in calculus courses.
Understanding Derivatives
A derivative in calculus represents the rate of change of a function with respect to a variable. Imagine the function as a curve on a graph; the derivative tells us how steep that curve is at any point.
  • It helps in understanding the behavior of functions, such as when they are increasing or decreasing.
  • In practical terms, a derivative allows you to find the slope of the tangent line to the function at a given point.

For the function \( y = x^n \), the derivative \( \frac{dy}{dx} = n \cdot x^{n-1} \) gives us precisely that slope form, where changes in \( x \) lead to changes in \( y \). Differentiating is fundamental in not just mathematics, but also physics, engineering, and other fields where modeling real-world systems is necessary.
Role of Exponents in Derivatives
Exponents in calculus play a crucial part when differentiating functions using the power rule. Here's how exponents influence differentiation:
  • They determine the nature of the function's growth or decay.
  • When differentiating using the power rule, the original exponent becomes a multiplier, effectively scaling the function.
  • The exponent is then decreased by one, changing the order of the term.

For example, with \( y = x^n \), the exponent "\( n \)" in \( x^n \) dictates that as you perform the differentiation, the resulting expression \( n \cdot x^{n-1} \) reflects a new relationship. This reflection demonstrates how the function's rate of change behaves as the values of \( x \) change, providing critical insights into the function's characteristics.

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Most popular questions from this chapter

Show that if a function \(f\) is defined on an interval symmetric about the origin (so that \(f\) is defined at \(-x\) whenever it is defined at \(x ),\) then $$f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$$ Then show that \((f(x)+f(-x)) / 2\) is even and that \((f(x)-\) \(f(-x) ) / 2\) is odd.

Evaluate the integrals in Exercises \(67-80\) $$ \int \frac{d x}{\sqrt{2 x-x^{2}}} $$

Skydiving If a body of mass \(m\) falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body's velocity \(t\) sec into the fall satisfies the differential equation $$m \frac{d v}{d t}=m g-k v^{2}$$ where \(k\) is a constant that depends on the body's aerodynamic properties and the density of the air. (We assume that the fall is short enough so that the variation in the air's density will not affect the outcome significantly.) a. Show that $$v=\sqrt{\frac{m g}{k}} \tanh \left(\sqrt{\frac{g k}{m}} t\right)$$satisfies the differential equation and the initial condition that\(v=0\) when \(t=0 .\) b. Find the body's limiting velocity, lim_{t\rightarrow\infty} v c. For a 160 -lb skydiver \((m g=160),\) with time in seconds and distance in feet, a typical value for \(k\) is \(0.005 .\) What is the diver's limiting velocity?

Evaluate the integrals in Exercises \(41-60\) $$\int 4 \cosh (3 x-\ln 2) d x$$

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