Problem 4 - Calculus of a Single Variable

Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ f(x)=-9 $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = -9\) is 0.

Step 1: Identify the Function

We first identify the function which is \(f(x) = -9\). This is a constant function where the output value is always -9, irrespective of the input value x.

Step 2: Apply Derivative Rules to the Constant Function

The rule of differentiation states that the derivative of a constant is always zero. Here, -9 is a constant. Therefore, its derivative is zero.

Step 3: Write the Final Answer

Hence, the derivate of the function \(f(x) = -9\) is 0, which can be represented as \(f'(x) = 0\).

Key Concepts

Derivative of a Constant

The derivative of a constant is a fundamental concept in calculus, which states that the derivative of any constant value is zero. When we talk about a constant, we mean a number that does not change regardless of the variable's value it is associated with. For example, in the expression \(f(x) = -9\), the number \(-9\) is a constant.
The reason the derivative of a constant is zero is that the rate of change of a constant value with respect to the variable, often x, is always zero.
Essentially:

• The derivative measures how a function's output changes as its input changes.
• A constant doesn't change, hence its rate of change, or derivative, is zero.

This understanding lays groundwork for approaching more complex differentiation tasks.

Rules of Differentiation

Rules of differentiation help us to find derivatives of functions systematically and efficiently. The basic rules include:

• Constant Rule: The derivative of a constant \(c\) is zero.
• Power Rule: For \(f(x) = x^n\), the derivative \(f'(x)\) is \(nx^{n-1}\).
• Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
• Product Rule: For two functions \(u(x)\) and \(v(x)\), \( (uv)' = u'v + uv' \).
• Quotient Rule: For \( g(x) = \frac{u(x)}{v(x)} \), \( g'(x) = \frac{u'v - uv'}{v^2} \).
• Chain Rule: For composite functions, the derivative is the derivative of the outer function times the derivative of the inner function.

These rules form the basis for calculating derivatives across a wide array of mathematical functions.

Constant Function Differentiation

Constant function differentiation is quite straightforward because it directly applies the constant rule of differentiation. In a constant function, such as \(f(x) = -9\), the output remains consistent no matter the input value.
When asked to differentiate a constant function:

• Identify the constant function, which is obvious when the function doesn't include variables, like \(x\).
• Apply the constant rule of differentiation: the derivative of \(f(x) = c\) is 0.
• The result, \(f'(x) = 0\), indicates that there is no change in the output value relative to change in input value.

This concept is prevalent in calculus and is an essential step in ensuring that students grasp the simplicity involved in differentiating constant functions. It sets a foundation for tackling more intricate problems efficiently.