Chapter 33: Problem 5
Find the derivative of \(y=8\)
Short Answer
Expert verified
The derivative of y=8 is 0.
Step by step solution
01
Understand the Problem
You are asked to find the derivative of the function defined as a constant, namely, \( y = 8 \). Understanding that \( y \) is constant is key to choosing the correct method of differentiation.
02
Identify the Rule for Differentiating Constants
In calculus, the derivative of a constant function \( c \) is 0. This is because a constant function does not change, and the derivative represents the rate of change of a function.
03
Apply the Derivative Rule for Constants
Using the rule that the derivative of a constant is zero, apply it to our function \( y = 8 \). This means \( \frac{dy}{dx} = 0 \).
04
Conclusion
The derivative of \( y = 8 \), a constant function, is 0. Thus, \( \frac{dy}{dx} = 0 \) accurately describes the rate of change of the function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivative of a Constant
In calculus, a fundamental concept is the derivative of a constant function. A constant function is a function that does not change, no matter the input value. Consider the function \( y = 8 \). Here, \( y \) is always 8, no matter what the input \( x \) is.
Plotting \( y = 8 \) results in a horizontal line on a graph. This visual representation shows that there is no change in \( y \) — it's always flat. Therefore, the derivative, which measures the rate of change, is zero.
Plotting \( y = 8 \) results in a horizontal line on a graph. This visual representation shows that there is no change in \( y \) — it's always flat. Therefore, the derivative, which measures the rate of change, is zero.
- Constant function has zero slope.
- Geometrically, this appears as a flat, horizontal line.
- The derivative \( \frac{dy}{dx} = 0 \) reflects no change, no slope, just constancy.
Rate of Change
The concept of rate of change in calculus refers to how one quantity changes with respect to another. It is pivotal because it defines the derivative of functions more broadly. The rate of change can be constant, as in our example of \( y = 8 \), or it can vary.
For joints that are not constants, the rate might increase or decrease. For any function \( f(x) \), the derivative \( \frac{df}{dx} \) represents how \( f \) changes when \( x \) changes.
For joints that are not constants, the rate might increase or decrease. For any function \( f(x) \), the derivative \( \frac{df}{dx} \) represents how \( f \) changes when \( x \) changes.
- Rate of change is key for understanding motion or growth.
- It helps predict future trends by looking at current behavior.
- Constant rate means no change, leading to a zero derivative.
Differentiation Rules
Differentiation rules are the methods applied to find the derivative of functions. Knowing the correct rule to apply is essential for solving calculus problems efficiently.
For example, constants use a simple rule: their derivative is always zero. This is just one of the many rules available. Others include the power rule, product rule, quotient rule, and chain rule. Each has specific applications based on the type of function.
For example, constants use a simple rule: their derivative is always zero. This is just one of the many rules available. Others include the power rule, product rule, quotient rule, and chain rule. Each has specific applications based on the type of function.
- Power Rule: Used for terms with exponents.
- Product Rule: Helps differentiate products of functions.
- Chain Rule: Essential for composite functions.