Question

True or False? In Exercises \(87-92,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ f(x)=g(x)+c, \text { then } f^{\prime}(x)=g^{\prime}(x) $$

Short answer

The statement is true.

Step-by-step solution

Identify the Given Function

The problem provides two functions. Function \(f(x)\) is equal to function \(g(x)\) plus a constant \(c\). So, we need to determine if the derivative of \(f(x)\) will be the same as the derivative of \(g(x)\).

Apply the Rule of Differentiation

When differentiating function \(f(x)\), we use the rule that the derivative of a constant is zero. This is a fundamental concept in calculus. So \(f'(x)\) is the derivative of \(g(x)\) plus the derivative of the constant \(c\). Since the derivative of a constant is zero, this leaves us with \(f'(x) = g'(x)\).

Verify the Outcome

Upon comparing the derivative of the given function \(f(x)\) and the derivative of function \(g(x)\), we find them to be the same. This confirms that the statement is true.

Key concepts

Differentiation Rules

Differentiation is a technique in calculus used to find the rate at which a function is changing at any given point. A set of rules helps in simplifying and solving differentiation problems. These are known as differentiation rules. Understanding these rules is essential for tackling more complex derivatives. Some of the basic differentiation rules include:

• Power Rule: Used primarily for polynomials, it states that the derivative of \(x^n\) is \(nx^{n-1}\).
• Product Rule: If you have a function that is the product of two functions, say \(u(x)\) and \(v(x)\), then its derivative is \(u'(x)v(x) + u(x)v'(x)\).
• Quotient Rule: For a function that is a quotient of two functions, \(u(x)/v(x)\), the derivative is \((u'(x)v(x) - u(x)v'(x))/v^2(x)\).
• Chain Rule: Useful for composite functions, it states that the derivative of \(f(g(x))\) is \(f'(g(x))g'(x)\).

These rules form the backbone of calculus differentiation. By mastering them, you can solve a wide array of calculus problems.

Derivative of a Constant

Calculus often deals with how functions change, but what about constants? Constants are numbers that do not change when a variable changes. The derivative of a constant is always zero. This is because a constant doesn’t change regardless of variations in the variable, showing a perfectly horizontal line if plotted on a graph.For instance, if \(c\) is a constant, then its derivative, denoted by \(\frac{d}{dx}(c)\), is zero: \(\frac{d}{dx}(c) = 0\). This is why when differentiating functions like \(f(x) = g(x) + c\), where \(c\) is a constant, the constant doesn't affect the rate of change of the original function \(g(x)\), thus leaving \(f'(x) = g'(x)\). This fundamental understanding helps simplify problems and verify the correctness of differentiation outcomes.

Function Derivatives

A function's derivative represents how the output value of a function changes as the input changes. In calculus, the derivative is often denoted by \(f'(x)\) or \(\frac{df}{dx}\). Knowing how to find function derivatives is vital since it gives insights into the behavior of functions.Common functions and their derivatives include:

• Constant function: As mentioned, derivatives are zero.
• Polynomial functions: Use the power rule to find derivatives.
• Exponential functions: The derivative of \(e^x\) is \(e^x\) itself.
• Trigonometric functions: For example, the derivative of \(\sin(x)\) is \(\cos(x)\), while the derivative of \(\cos(x)\) is \(-\sin(x)\).

Differentiation of functions explains a function's behavior and predicts how small changes can impact the overall output. This makes derivatives one of the most powerful tools in mathematics.