Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(g(x)=3 f(x),\) then \(g^{\prime}(x)=3 f^{\prime}(x)\)

Short answer

The statement is true.

Step-by-step solution

Apply the Derivative Rule

The derivative of a constant times a function is the constant times the derivative of the function. Therefore, when we differentiate \(g(x) = 3f(x)\), applying the Constants Rule of derivatives, we would have \(g'(x) = 3f'(x)\).

Comparison and Conclusion

Comparing the obtained derivative of \(g(x)\) with \(3f'(x)\), we can see that they match. This verifies the statement, leading us to conclude that the statement is true.

Key concepts

Constants Rule of Derivatives

Understanding the Constants Rule of Derivatives is a cornerstone in calculus, particularly when dealing with differentiation. We often come across functions that are multiplied by a constant. The Constants Rule simplifies the process of differentiation, stating that the derivative of a constant multiplied by a function is simply the constant multiplied by the derivative of the function.

For example, if you have a function like \(g(x) = c \times f(x)\), where \(c\) is a constant, the derivative \(g'(x)\) is computed as \(c \times f'(x)\). This principle allows us to 'pull out' the constant and focus on differentiating the function itself. It's essential to remember that this rule only applies to constants; variables would need to be managed using other differentiation rules.

When we apply this rule to our original exercise, where \(g(x) = 3f(x)\), we differentiate \(f(x)\) normally and then multiply by 3, resulting in \(g'(x) = 3f'(x)\), just as the solution outlined. The beauty of the Constants Rule is its simplicity and versatility, making it a valuable tool in the arsenal of calculus techniques.

Differentiation

Differentiation is one of the fundamental operations in calculus, useful for finding the rate at which a quantity changes. It essentially measures the sensitivity to change of a function value (output) with respect to a change in its argument (input).

The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. When we differentiated the function \(g(x) = 3f(x)\), what we were actually calculating was how much \(g(x)\) changes as \(x\) changes, keeping in mind the multiplication by 3 as a scaling factor due to the Constants Rule.

Significance of Differentiation

Diffrentiation is not only a mechanical process but it has broad applications in physics, engineering, economics, and beyond. For instance, it helps in determining speed from a position-time graph, finding the marginal cost or revenue in economics, and even computing the optimal dimensions of a structure in engineering.

Calculus Principles

Calculus is a branch of mathematics that deals with the properties and applications of derivatives and integrals. Its principles are fundamental in understanding continuous change and are divided primarily into differential calculus and integral calculus.

Differential calculus, as employed in our exercise example, concerns the concept of instantaneous rate of change, and is used to find the slope of curves. Integral calculus, on the other hand, is centered around accumulation of quantities, such as areas under curves. Both of these concepts are interrelated through the Fundamental Theorem of Calculus, which links the concept of differentiation with that of integration.

Core Calculus Concepts

• Limit: The value that a function 'approaches' as the input 'approaches' some value.
• Continuity: A property where a function has no interruptions or 'jumps' in its graph.
• Derivative: Measures the rate of change of a function with respect to a variable.
• Integral: Represents the accumulation of quantities and can find the area under a curve.

Understanding these principles is essential for solving problems in calculus, much like how we successfully applied the Constants Rule of Derivatives to validate the matching statement in the exercise.