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Chapter 3: Problem 6

Find \(D_{x} y\) using the rules of this section. $$ y=-3 x^{-4} $$

Short Answer

Expert verified
The derivative is \( 12x^{-5} \).

Step by step solution

01

Identify the function form

The given function is in the form of a power function: \( y = -3x^{-4} \). The goal is to find the derivative of this function with respect to \( x \), which we'll denote as \( D_x y \) or \( \frac{dy}{dx} \).
02

Apply the power rule for differentiation

The power rule for differentiation states that the derivative of \( ax^n \) is \( anx^{n-1} \). Here, \( a = -3 \) and \( n = -4 \). Applying the power rule gives:\[ \frac{dy}{dx} = -3 \cdot (-4) \cdot x^{-4-1} \]
03

Simplify the expression

Carry out the multiplication and simplification:\[ \frac{dy}{dx} = 12x^{-5} \]Thus, the derivative \( D_x y \) is \( 12x^{-5} \).

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Key Concepts

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

Power Rule
The power rule is a fundamental concept in calculus that simplifies finding the derivative of a power function. It's particularly handy because many functions can be expressed in terms of powers of variables. The rule is simple yet powerful:
  • If you have a function in the form of \( ax^n \), where \( a \) and \( n \) are constants, the derivative is \( anx^{n-1} \).
  • For example, the derivative of \( y = -3x^{-4} \) is found by identifying \( a = -3 \) and \( n = -4 \), and then applying the power rule.
When you apply the power rule, multiply the coefficient \( a \) by the exponent \( n \), and decrease the exponent by one. This method saves time and is often the first step in the differentiation process, making it an essential tool in your calculus toolkit.
Differentiation
Differentiation is the process of finding the derivative of a function. It's a cornerstone of calculus and helps us understand how functions change. In simpler terms, differentiation tells us the function's rate of change at any point.
  • The derivative is denoted as \( D_x y \) or \( \frac{dy}{dx} \).
  • In our example, we used the power rule to differentiate \( y = -3x^{-4} \), resulting in the derivative function \( 12x^{-5} \).
Finding the derivative allows us to answer questions about the slope of a tangent line at any point on a curve, and whether the function is increasing or decreasing at that point. Differentiation is also crucial for optimization problems and in physics to find velocity and acceleration from displacement-time functions.
Calculus
Calculus is a branch of mathematics that studies how things change. It's divided into two main areas: differentiation and integration. Together, they provide tools to model and analyze dynamic systems in real life, from natural sciences to economics.
  • In calculus, differentiation lets us find the instantaneous rate of change of a function, which is essential in various applications.
  • The process we discussed, using the power rule to find \( D_x y \) for \( y = -3x^{-4} \), is an application of differential calculus.
Through calculus, we can solve complex problems by breaking them down into simpler parts, analyzing the rate of change, and calculating accumulated quantities. It's a wide world that spans from finding the area under curves to predicting the future trajectory of a moving object.

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Most popular questions from this chapter

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ D_{x} \sec ^{3} F(x) $$

, find dy/dx by logarithmic differentiation. $$ y=\frac{\left(x^{2}+3\right)^{2 / 3}(3 x+2)^{2}}{\sqrt{x+1}} $$

If \(y=x^{2}-3\), find the values of \(\Delta y\) and \(d y\) in each case. (a) \(x=2\) and \(d x=\Delta x=0.5\) (b) \(x=3\) and \(d x=\Delta x=-0.12\)

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ h(x)=x \sec x \text { at } a=0,(-\pi / 2, \pi / 2) $$

$$ \text { } , \text { find the indicated derivative. } $$ $$ D_{\theta} \sqrt{\log _{10}\left(3^{\theta^{2}-\theta}\right)} $$
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