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Chapter 1: Problem 23

Find \(\frac{d y}{d x}\). $$y=\frac{4 x}{5}$$

Short Answer

Expert verified
The derivative of \( y = \frac{4x}{5} \) is \( \frac{dy}{dx} = \frac{4}{5} \)

Step by step solution

01

Identify the Function

The given function is \(y = \frac{4x}{5}\). This is a simple linear function with a constant coefficient.
02

Differentiate the Function

To find \( \frac{dy}{dx} \), apply the power rule of differentiation to each term of the function. The differentiation of a constant multiplied by \(x\) is the constant. Therefore, \( \frac{d}{dx} \left( \frac{4x}{5} \right) = \frac{4}{5} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Function
A linear function is one of the most basic types of functions in mathematics. It takes the form of y = mx + b, where 'm' is the slope and 'b' is the y-intercept. These functions graph as straight lines on the coordinate plane.
In our exercise, the given function is y = \(\frac{4x}{5}\), which is equivalent to \(\frac{4}{5}x + 0\). Here, the slope 'm' is \(\frac{4}{5}\) and the y-intercept 'b' is 0. The simplicity of linear functions makes differentiating them straightforward.
Power Rule
The power rule is a fundamental tool in differentiation, allowing you to find the derivative of functions of the form \(x^n\). It states:
If y = \(x^n\), then \(\frac{d y}{dx} = n x^{(n-1)}\).
However, our exercise deals with a linear function: y = \(\frac{4x}{5}\), which can be seen as \(4/5x^1\). Applying the power rule for n=1:
y = \(4/5 x^1\)
\(\frac{dy}{dx} = 1 (4/5)x^{1-1}\)
which simplifies to \(\frac{4}{5}\). This step-by-step process shows how the power rule was applied.
Constant Coefficient
In the context of differentiation, a constant coefficient is a number that multiplies a variable. This number remains unchanged when differentiating.
In the function y = \(\frac{4x}{5}\), the constant coefficient is \(\frac{4}{5}\). When differentiating, the constant coefficient is preserved:
\(\frac{d}{dx}\left(\frac{4}{5} x\right) = \frac{4}{5}\). This result is directly derived from the linearity of differentiation and the power rule.

When you encounter a linear function with a constant coefficient, remember that the derivative will always be the coefficient itself.

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Most popular questions from this chapter

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