Problem 12
Question
Find \(\frac{d y}{d x}\). $$ y=4 x^{-2} $$
Step-by-Step Solution
Verified Answer
\( \frac{dy}{dx} = -8x^{-3}\).
1Step 1: Identify the Differentiation Rule
The function given is in the form of a power of x: \(y = 4x^{-2}\). We can apply the power rule for differentiation, which states that if \(y = ax^n\), then \( \frac{dy}{dx} = a n x^{n-1} \).
2Step 2: Apply the Power Rule
From the expression \(y = 4x^{-2}\), identify \(a = 4\) and \(n = -2\). According to the power rule, the derivative is given by \( \frac{dy}{dx} = 4(-2)x^{-3}\).
3Step 3: Simplify the Expression
Simplify the expression \( \frac{dy}{dx} = 4(-2)x^{-3}\) to get \( \frac{dy}{dx} = -8x^{-3}\).
Key Concepts
Power RuleDerivative of Power FunctionsBasic Differentiation Rules
Power Rule
In calculus, the power rule is a basic differentiation technique used to find the derivative of functions in the form of a power of x. For a function presented as \( y = ax^n \), the power rule states that the derivative \( \frac{dy}{dx} \) is obtained by multiplying the exponent \( n \) by the coefficient \( a \), then reducing the exponent by one, resulting in \( \frac{dy}{dx} = a n x^{n-1} \).
This rule is incredibly useful because it simplifies the differentiation of polynomial functions, which appear frequently in calculus. The power rule applies to both positive and negative powers, as well as fractional powers of x. This makes it very flexible and widely applicable across different calculus problems.
This rule is incredibly useful because it simplifies the differentiation of polynomial functions, which appear frequently in calculus. The power rule applies to both positive and negative powers, as well as fractional powers of x. This makes it very flexible and widely applicable across different calculus problems.
Derivative of Power Functions
Understanding the derivative of power functions means grasping how the power rule is applied to functions expressed as \( y = ax^n \). A power function includes any expression where a constant multiplier \( a \) is combined with \( x \) raised to a power \( n \).
When differentiating these functions, regardless of whether \( n \) is positive, negative, or fractional, the process remains consistent with applying the power rule.
For example, in differentiating \( y = 4x^{-2} \), we identify \( n = -2 \), so the derivative becomes \( \frac{dy}{dx} = 4(-2)x^{-3} \), simplifying to \( -8x^{-3} \).
When the exponent is negative, the process still follows the same steps as with positive exponents, reinforcing the versatility of the power rule.
When differentiating these functions, regardless of whether \( n \) is positive, negative, or fractional, the process remains consistent with applying the power rule.
- Multiply the original exponent \( n \) by the coefficient \( a \).
- Reduce the exponent by 1 to form the new power of x.
For example, in differentiating \( y = 4x^{-2} \), we identify \( n = -2 \), so the derivative becomes \( \frac{dy}{dx} = 4(-2)x^{-3} \), simplifying to \( -8x^{-3} \).
When the exponent is negative, the process still follows the same steps as with positive exponents, reinforcing the versatility of the power rule.
Basic Differentiation Rules
Basic differentiation rules are fundamental tools in calculus that help in finding the rate of change of functions. Apart from the power rule, there are several other rules used to differentiate functions with different properties and combinations:
These rules complement the power rule, allowing for a comprehensive approach to differentiating a wide variety of functions, whether they are simple polynomials or more complex expressions involving combinations of multiple terms.
- Constant Rule: The derivative of a constant is zero because a constant does not change.
- Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
- Product Rule: Use this rule when differentiating products of two functions.
- Quotient Rule: Useful for finding the derivative of a quotient of two functions.
- Chain Rule: This rule is invaluable for differentiating composite functions where one function is inside another.
These rules complement the power rule, allowing for a comprehensive approach to differentiating a wide variety of functions, whether they are simple polynomials or more complex expressions involving combinations of multiple terms.
Other exercises in this chapter
Problem 11
Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow-1}\left(x^{2}-4
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