Question

Find the derivative of \(y=8\)

Short answer

The derivative of y=8 is 0.

Step-by-step solution

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.

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.

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 \( \frac{dy}{dx} = 0 \).

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.

Key concepts

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.

• 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.

In essence, if a function doesn't change, its rate of change is zero. This is why the derivative of any constant is 0.

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.

• 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.

The notion aids in comprehending dynamic systems and other real-world situations where change is inevitable, unlike our constant example.

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.

• Power Rule: Used for terms with exponents.
• Product Rule: Helps differentiate products of functions.
• Chain Rule: Essential for composite functions.

Mastery of these rules is vital for success in calculus, allowing you to tackle a wide range of functions accurately and swiftly.