# I applying the difference/power rule (derivatives)

• Sentience
In summary, the conversation is about finding the derivative of the expression (x - a) using the Power Rule and Difference Rule. The student is confused about the answer being 1 and believes it should be 0 because both variables are to the power of 1. The tutor clarifies that 'a' is a constant and the derivative of a constant is 0. The student is advised to think of the function graphically to understand this concept better.
Sentience

## Homework Statement

The problem is : take the derivative of (x - a)

## Homework Equations

Power Rule : f '(x) = r x^(r-1)

Difference Rule : f '(x) = g '(x) - h '(x)

## The Attempt at a Solution

This is such a simple problem but I don't understand how my solutions manual and Wolfram Alpha came to the answer. According to these sources the answer is 1.

However, because both variables in the expression are to a single power I was under the impression the derivative of each variable would equal 1, leading to (1 - 1) = 0

If you need any clarification, please let me know.

The parameter 'a' represents a constant, not a variable. What's the derivative of a constant?

Keep in mind, x = x^1, so you can apply the power rule for this.

If it helps, try to graph the function in your head. For instance if you have a constant, $$\beta$$, which is a straight line intersecting the y-axis. Then it becomes clear that taking the derivative or finding another function that gives the slope at each point along $$\beta$$ would definitely be $$0$$ because a horizontal line has no slope.

## What is the difference/power rule for derivatives?

The difference/power rule is a formula used to find the derivative of a function that is the difference of two other functions or has a power function raised to a constant power. It states that the derivative of a difference of two functions is equal to the difference of their derivatives, and the derivative of a power function is equal to the constant power multiplied by the derivative of the base function.

## How is the difference/power rule used in calculus?

The difference/power rule is a fundamental concept in calculus that allows us to find the rate of change of a function at any given point. It is used to find derivatives of functions that are not easily differentiable using other methods, making it an essential tool in many applications of calculus.

## What are the steps for applying the difference/power rule?

The steps for applying the difference/power rule are as follows:
1. Identify the function and rewrite it in the form of (f(x)-g(x)) or (f(x))^n.
2. Apply the rule by finding the individual derivatives of f(x) and g(x) or the derivative of f(x) with respect to x and multiplying it by the constant power n.
3. Simplify the derivative by combining like terms and removing any unnecessary parentheses.

## Are there any exceptions or special cases for the difference/power rule?

Yes, there are some exceptions and special cases for the difference/power rule. For example, if the function has a power function raised to a variable power, the rule cannot be directly applied, and other methods must be used. Additionally, the rule does not apply to functions with logarithmic or trigonometric functions.

## How can the difference/power rule be used in real-life applications?

The difference/power rule has many real-life applications, particularly in fields such as physics, engineering, and economics. It can be used to find the velocity and acceleration of moving objects, the rate of change of temperature in a chemical reaction, and the marginal cost of production in economics. It is also used in optimization problems to find the maximum or minimum value of a function.

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