Problem 1 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=5$$

Step-by-Step Solution

Answer

The derivative of \(y = 5\) is 0.

Step 1: Understand the Constant Derivative Rule

The derivative of a constant function is always zero. This rule is based on the concept that a constant function does not change its value regardless of its input, hence its rate of change (derivative) is zero.

Step 2: Identify the Constant Function

In this problem, the given function is \(y = 5\). Since 5 is a constant, the question simply asks for the differentiation of a constant function.

Step 3: Apply the Constant Derivative Rule

Using the rule explained in Step 1, the derivative of the constant function \(y = 5\) with respect to any variable (typically \(x\)) is 0.

Key Concepts

Derivative

In calculus, the derivative is a fundamental concept that represents an instantaneous rate of change. When you have a function, the derivative tells you how the function's output value changes as its input changes. This is much like how speed tells you how your position changes over time.

Derivatives are crucial in a wide range of applications:

• In physics, derivatives are used to compute velocity and acceleration.
• In economics, they can help find marginal cost and revenue.
• In engineering, they are essential for understanding rates of change in systems.

Mathematically, the derivative at a point can be thought of as the slope of the tangent line to the curve at that point. For a function \(f(x)\), the derivative is often written as \(f'(x)\) or \(\frac{df}{dx}\). This notation indicates the specific rate at which \(f\) is changing at any given value of \(x\).

Constant Function

A constant function is a type of function where the output value remains the same, no matter what the input is. For example, if you have a function \(f(x) = c\), where \(c\) is a constant, no matter how \(x\) changes, \(f(x)\) remains equal to \(c\).

Characteristics of constant functions include:

• The graph of a constant function is a horizontal line.
• There is no dependence on the variable; changing the input does not change the output.

When taking the derivative of a constant function, like in the original exercise where the function is \(y = 5\), the result is always 0. This is because there is zero change in the output as the input varies, meaning the rate of change is nonexistent.

Rate of Change

The rate of change is a key idea in understanding derivatives and what they represent. Essentially, it tells us how one quantity changes in relation to another. This concept is particularly highlighted in calculus through the derivative.

Here are some key points about rate of change:

• It can be constant or varying, depending on the function.
• A derivative gives the exact rate of change at any point for a given function.
• For linear functions, the rate of change is constant, while for nonlinear functions, it can vary across different intervals.

For the function \(y = 5\) given in the exercise, the rate of change is zero. This means that no matter how you change the value of \(x\), \(y\) does not change at all. Therefore, the derivative is 0, indicating no change in the output per change in input.