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Magazine Chapter 4: Problem 60
Find \(\frac{d y}{d x}\) $$ y=x^{-2}+3 x^{-4} $$
Short Answer
Expert verified
The derivative is \(-2x^{-3} - 12x^{-5}\).
Step by step solution
01
Identify the function components
In the given function, identify each term that needs to be differentiated. The function provided is \(y = x^{-2} + 3x^{-4}\), which consists of two terms: \(x^{-2}\) and \(3x^{-4}\).
02
Differentiate the first term
Apply the power rule of differentiation to the first term \(x^{-2}\). The power rule states that \(\frac{d}{dx}(x^n) = nx^{n-1}\). Applying it to \(x^{-2}\), we get \(-2x^{-3}\).
03
Differentiate the second term
Now, differentiate the second term \(3x^{-4}\) also using the power rule. Here, \(n = -4\), so \(\frac{d}{dx}(3x^{-4}) = 3 \cdot (-4) x^{-5} = -12x^{-5}\).
04
Combine the derivatives
Combine the derivatives of both terms to obtain the final derivative of the function. Thus, \(\frac{dy}{dx} = -2x^{-3} - 12x^{-5}\).
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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 a crucial tool for differentiation, enabling us to find the derivative of polynomial terms quickly and efficiently. The rule states that if you have a function of the form \( f(x) = x^n \), then the derivative with respect to \( x \) is \( \frac{d}{dx}(x^n) = nx^{n-1} \).
This means you take the exponent \( n \), multiply it by the coefficient in front of \( x \) (if it exists), and then subtract one from the exponent. This rule simplifies the process of finding a derivative, especially for functions consisting of multiple polynomial terms.
This means you take the exponent \( n \), multiply it by the coefficient in front of \( x \) (if it exists), and then subtract one from the exponent. This rule simplifies the process of finding a derivative, especially for functions consisting of multiple polynomial terms.
- For example, applying the power rule to \( x^{-2} \) gives \( -2x^{-3} \). This is because you multiply \(-2\) by one (implicit coefficient) and decrease the exponent by one.
- Similarly, for \( 3x^{-4} \), applying the power rule results in \(-12x^{-5}\). Multiply \(-4\) by the coefficient 3, then subtract one from the exponent.
Functions
Functions are the building blocks in calculus and mathematics in general. A function assigns a unique output to every input of a given set. In our exercise, the function is given by \( y = x^{-2} + 3x^{-4} \). This function is composed of two terms: \( x^{-2} \) and \( 3x^{-4} \), each of which needs to be differentiated separately.
When working with functions:
When working with functions:
- Identify each component of the function that will be differentiated. This makes it easier to apply differentiation rules, like the power rule, to each part.
- Always keep in mind the overall structure of the function 鈥� even when it's broken down into terms, it still represents a whole, cohesive mathematical expression.
Calculus Problem Solving
Solving calculus problems often requires a strategic approach. In our exercise, the task is to find the derivative of \( y = x^{-2} + 3x^{-4} \). Here is a step-by-step strategy for solving such problems:
- **Identify the Problem:** First, clearly define what needs to be solved. Here, it involves finding the derivative \( \frac{dy}{dx} \) of the given function.
- **Break It Down:** Divide the problem into manageable parts. Separate the function into individual terms, each needing differentiation.
- **Apply the Rules:** Use appropriate differentiation rules. The power rule is applied to each term separately in this instance.
- **Combine Your Work:** After differentiating each term, combine the results to find the overall derivative. For our problem, this means adding the derivatives from both terms to form \( \frac{dy}{dx} = -2x^{-3} - 12x^{-5} \).
