Problem 6 - Calculus

Question

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

Step-by-Step Solution

Answer

The derivative is \( D_x y = 12x^{-5} \).

Step 1: Identify the form of the function

The given function is \( y = -3x^{-4} \). This function is in the form \( ax^n \), where \( a = -3 \) and \( n = -4 \).

Step 2: Apply the power rule for differentiation

The power rule for differentiation states that if \( y = ax^n \), then \( D_x y = nax^{n-1} \). We need to apply this rule to our function.

Step 3: Differentiate the function

Using the power rule, differentiate the function: \[ D_x y = (-4)(-3)x^{-4-1} = 12x^{-5} \].

Step 4: Write the derivative

The derivative of the function \( y = -3x^{-4} \) with respect to \( x \) is \( D_x y = 12x^{-5} \).

Key Concepts

Power Rule

The power rule is one of the simplest and most widely used rules in calculus to find derivatives. It comes in handy when differentiating algebraic functions, especially those of the form \( ax^n \). Here’s how the power rule works:

• If you have a function \( y = ax^n \), where \( a \) is a constant and \( n \) is a real number, the derivative of \( y \) with respect to \( x \) is found by multiplying \( a \) by the exponent \( n \), and then decreasing the exponent by one.
• In formula terms, \( D_x y = nax^{n-1} \).

Applying the power rule is straightforward. For instance, if your function is \( y = -3x^{-4} \), identify \( a = -3 \) and \( n = -4 \). Plug these values into the rule to calculate the derivative, resulting in \( 12x^{-5} \). Notice how we lowered the exponent from \(-4\) to \(-5\) after multiplying by the exponent itself.

Derivative

A derivative is a fundamental concept in calculus that measures how a function changes as its input changes. In algebraic terms, it represents the rate of change or the slope of the function at any given point. The derivative is denoted by \( D_x y \) or \( y' \) when dealing with a function \( y = f(x) \).In practical terms, the derivative analyzes how a small change in \( x \) results in a change in \( y \). Think of it like this: if you know the derivative, you can predict how the function behaves with small changes in its input.When the derivative of \( y = -3x^{-4} \) is \( D_x y = 12x^{-5} \), it tells us that for every small change in \( x \), \( y \) changes proportionally by \( 12x^{-5} \). The sign and the size of the derivative give insights into the function’s increasing or decreasing nature and its steepness.

Algebraic Function Differentiation

Algebraic function differentiation involves finding the derivative of functions composed of algebraic terms such as constants, variables, and exponents. When differentiating such functions, basic rules of differentiation, like the power rule, are applied. This process plays a key role in calculus, as it helps in understanding the behavior and properties of polynomial equations.Here’s what you need to keep in mind when differentiating algebraic functions:

• Break down the function to identify each term's base and exponent.
• Apply relevant differentiation rules such as the power rule.
• Simplify the resulting expression, looking out for negative and zero exponents which indicate the function's behavior.

In our example, \( y = -3x^{-4} \), applying the power rule gives us \( D_x y = 12x^{-5} \). Differentiating even seemingly complex algebraic functions becomes approachable when you recognize the terms and apply the appropriate rules step by step. Ultimately, mastering these simple techniques allows for enhanced understanding and solving of broader calculus problems.