Problem 18 - Calculus and its applications

Question

Find \(\frac{d y}{d x}\). $$ y=x^{0.9} $$

Step-by-Step Solution

Answer

\( \frac{d y}{dx} = 0.9 \cdot x^{-0.1} \)

Step 1: Identify the Power Rule

First, recognize that the function given is in the form of a power: \( y = x^{n} \), where \( n = 0.9 \). The power rule for derivatives is \( \frac{d}{dx} [x^n] = n \cdot x^{n-1} \).

Step 2: Apply the Power Rule

Apply the power rule to differentiate \( y = x^{0.9} \). According to the rule, multiply the exponent by \( x \) raised to the power of one less than the original exponent. This gives: \( \frac{d y}{dx} = 0.9 \cdot x^{0.9-1} \).

Step 3: Simplify the Expression

Subtract 1 from the exponent in the expression \( x^{0.9 - 1} \) to simplify: \( x^{-0.1} \). Therefore, the derivative simplifies to \( \frac{d y}{dx} = 0.9 \cdot x^{-0.1} \).

Key Concepts

Understanding the Power Rule

The power rule is a fundamental tool in calculus, particularly when dealing with derivatives of polynomial expressions. When using the power rule, it's all about working with exponents. This rule allows us to differentiate functions of the form \( y = x^n \). The beauty of this rule lies in its simplicity. To differentiate, you multiply the original exponent, \( n \), by the base, \( x \), and reduce the exponent by one. So, for an expression like \( x^n \), the derivative is found using the formula: \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \).
This means you are scaling down the influence of the x-variable as the exponent decreases.
When applying it to our problem where \( y = x^{0.9} \), you can see how straightforward the rule becomes. You take 0.9 (the exponent) and multiply it by the next power level of \( x \), which is 0.9 - 1, or -0.1. This is how the derivative \( \frac{dy}{dx} = 0.9 \cdot x^{-0.1} \) is calculated.

What is Differentiation?

Differentiation is a process that lets us find out how a function changes when its input changes. It shows the rate of change and is commonly used to find slopes of curves. In essence, it's about understanding how the output of a function varies as the input changes.
For instance, in physical sciences, this can relate to speed or acceleration. Differentiation deals with limits and small changes over a gradual point, thus involving a form of precise calculus.

• This process can be guided by basic differentiation rules, such as the power rule.
• When applying it, you make the leap from a function to its rate of change function, often called the derivative.

For the function \( y = x^{0.9} \), differentiation using the power rule allows us to find \( \frac{dy}{dx} = 0.9 \cdot x^{-0.1} \), giving us a sense of how \( y \) behaves as \( x \) evolves.

Working with Exponent Rules

Exponent rules are essential for manipulating and understanding exponential expressions. These rules simplify expressions and can help with calculations, especially during differentiation.
There are a few key rules:

• Product Rule: \( x^a \cdot x^b = x^{a+b} \)
• Quotient Rule: \( \frac{x^a}{x^b} = x^{a-b} \)
• Power of a Power Rule: \((x^a)^b = x^{a \cdot b}\)

When differentiating \( x^{0.9} \), exponent rules come into play as you simplify the expression to \( x^{-0.1} \).
Subtracting one from the original exponent \( (0.9 - 1) = -0.1 \) uses the basic concept of subtracting exponents, which is crucial for simplifying derivatives.