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. $$ g(x)=3 f(x), \text { then } g^{\prime}(x)=3 f^{\prime}(x) $$

Short answer

The statement is true.

Step-by-step solution

Understanding the Constants with Derivatives

In the world of derivatives, constants work very differently than variables. When you take the derivative of a constant multiplied by a function, the constant stays the same and you only take the derivative of the function. This is because the rate of change of a constant is zero, and so the constant doesn't affect the derivative of the variable --> it is just multiplied by it.

Apply the rule to function g(x)

With the rule of constants, we can apply it to the question at hand. So if \( g(x)=3f(x) \), and we want to know \( g'(x) \), we leave the constant as it is, and take the derivative of function f(x) ---> \( g'(x)=3f'(x) \)

Compare the result with the given statement

Comparing the calculated derivative \( g'(x)=3f'(x) \) with the given statement \( g'(x)=3f'(x) \), we can see that they match exactly. It means the given statement is correct.

Key concepts

Constants in Derivatives

Constants play a unique role when taking the derivative of a function. When a constant is multiplied by a function, it is preserved in the process of differentiation. This means that the constant itself does not change; rather, it scales the rate of change of the function it multiplies. For example, if you have a function \( g(x) = c \cdot f(x) \), where \( c \) is a constant, the derivative \( g'(x) \) is computed as \( g'(x) = c \cdot f'(x) \).
This is a result of the linearity of differentiation, which makes it straightforward to handle constants. The constant "tags along" as you calculate the derivative of the function part, emphasizing that constants do not influence the core rate of change but rather the amplitude of it.
This important property is one of the foundational rules in calculus and helps ease understanding and calculation of derivatives during problem-solving.

Calculus Problem Solving

Tackling calculus problems often involves following a systematic approach. The goal is to deconstruct a problem and break it down into manageable steps. When dealing with derivatives, the following steps can be useful:

• Identify the given function and the specific operations involved, such as multiplication by a constant.
• Recall any relevant differentiation rules, like the constant multiple rule or sum and difference rules.
• Apply these rules step by step to derive the function correctly.
• Compare your resultant expression with any given statements to check accuracy.

Understanding and applying these steps ensures clarity and organization in calculus problem-solving. This approach reduces errors and builds confidence in handling even complex derivatives.
Remembering specific properties like how constants behave during differentiation can save time and streamline the process.

Function Differentiation

Differentiation is a fundamental concept in calculus, focusing on how functions change. It is the process of finding the derivative of a function, which represents the rate of change concerning its variable. For a function like \( f(x) \), its derivative \( f'(x) \) provides insights into how the function grows or declines as \( x \) varies.
Some critical points to remember when differentiating include:

• Constant functions: The derivative of a constant is zero.
• Power rule: For functions \( x^n \), \( \frac{d}{dx}x^n = nx^{n-1} \).
• Constant multiple rule: If \( g(x) = c \cdot f(x) \), then \( g'(x) = c \cdot f'(x) \).
• Sum and difference: The derivative of a sum/difference is the sum/difference of their derivatives.

Building a strong grasp of these principles facilitates effortless navigation of various calculus problems. Function differentiation lays the groundwork for more advanced topics, such as integration and differential equations.
Approaching differentiation with a clear understanding of its rules equips students with the skills to analyze and interpret the behavior of functions effectively.