Problem 17 - Calculus and its applications

Question

Find \(\frac{d y}{d x}\). $$ y=x^{0.7} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{d y}{d x} = 0.7 \cdot x^{-0.3} \).

Step 1: Understanding the Problem

We need to find the derivative of the function \( y = x^{0.7} \) with respect to \( x \). This requires applying the power rule for differentiation.

Step 2: Applying the Power Rule

The power rule for differentiation states that if \( y = x^n \), then \( \frac{d y}{d x} = n \cdot x^{n-1} \). In our case, \( n = 0.7 \). Thus, \( \frac{d y}{d x} = 0.7 \cdot x^{0.7 - 1} \).

Step 3: Simplifying the Expression

Compute \( 0.7 - 1 \) to simplify the exponent: \( 0.7 - 1 = -0.3 \). Therefore, the derivative simplifies to \( \frac{d y}{d x} = 0.7 \cdot x^{-0.3} \).

Key Concepts

Derivative

In calculus, the derivative of a function measures how the function's output value changes as its input changes. It's essentially a way of determining the rate of change or slope of a function at any given point. This is crucial in understanding how different variables interrelate. For example, if we consider the function of time and position, the derivative can tell us about the speed. Derivatives can be taken for simple functions, like polynomials, as well as more complex ones, by using specific rules for differentiation. In the exercise mentioned, finding the derivative of \( y = x^{0.7} \) helps us understand how \( y \) changes with small changes in \( x \). The derivative at any point gives a slope of the tangent line that touches the function graph at that point.

Power Rule

The power rule is one of the fundamental rules of differentiation used to find the derivative of a function of the form \( y = x^n \). This rule makes it significantly easier to calculate derivatives for powers of \( x \). Here's the general idea:

• Given a function \( y = x^n \), its derivative can be found using \( \frac{d y}{d x} = n \cdot x^{n-1} \).
• In simple terms, this means you bring the power (\( n \)) down as a coefficient and decrease the original power by one.
• For example, with the function \( y = x^{0.7} \), using the power rule, the derivative will be \( 0.7 \cdot x^{0.7-1} \), which simplifies to \( 0.7 \cdot x^{-0.3} \).

This rule proves very efficient, especially for polynomial functions, and is a key component of calculus, making it easier to handle complex problems strategically.

Differentiation

Differentiation is the process of finding the derivative of a function. It forms the core of calculus and involves applying a variety of rules and methods to determine how a function changes at any given point. When differentiating, it's crucial to select the appropriate rule that fits the function type you're dealing with. Here's an overview:

• The goal of differentiation is to find a derivative, which tells us the rate of change.
• Common rules for differentiation include the power rule, product rule, quotient rule, and chain rule.
• By applying these rules methodically, one can solve a range of problems from simple to complex.
• In our exercise, the differentiation of \( y = x^{0.7} \) was accomplished using the power rule.

Differentiation is not just a mathematical operation but a tool that allows us to explore and understand patterns and changes in different contexts, whether in physics, engineering, economics, or beyond.