Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } f(x)=g(x)+c, \text { then } f^{\prime}(x)=g^{\prime}(x) \text { . } $$

Short answer

The statement is true. The derivative of the function \(f(x) = g(x) + c\) is indeed \(f'(x) = g'(x)\), with the constant \(c\) having no impact on the derivative of the function.

Step-by-step solution

Understand the principles of Differentiation

The derivative of a function measures how the function changes as the variables change. It's important to remember that the derivative of a constant always equals zero. This is a fundamental principle of differentiation.

Write out the equation

The statement provided is \(f(x) = g(x) + c\), and we are trying to determine if \(f'(x) = g'(x)\). Considering the principles in step 1, we proceed to compute the derivative.

Compute the derivative of \(f(x)\)

We will differentiate \(f(x) = g(x) + c\). Applying the principles of differentiation, the derivative of \(g(x)\) is \(g'(x)\) and the derivative of a constant \(c\) is zero. Hence, \(f'(x) = g'(x) + 0\).

Conclusion

From Step 3, we have \(f'(x) = g'(x) + 0\). Since adding zero to a quantity does not change the quantity, we can simplify to \(f'(x) = g'(x)\). Therefore, the statement 'If \(f(x) = g(x) + c\), then \(f'(x) = g'(x)\)' is true.

Key concepts

Derivative of a Constant

Understanding the derivative of a constant is essential in calculus. When dealing with constant values in a function, it's crucial to remember this simple rule: the derivative of any constant is always zero.

Let's consider why that is. Differentiation measures the rate at which function values change with respect to changes in the input value. Since a constant does not change no matter how much the input changes, its rate of change is intuitively zero. Mathematically, if you have a constant function such as f(x) = c, where c is a constant, the derivative f'(x), according to differentiation rules, is 0.

This straightforward principle helps to simplify many problems in calculus. Whenever you have an equation involving constants in addition to other terms, you can confidently differentiate the constants to zero and focus on the variable terms.

Computing Derivatives

When it comes to computing derivatives, there are several rules and techniques that come into play. Differentiation is not just a mechanical process; understanding how it works can significantly improve your ability to solve calculus problems.

For example, consider the function f(x) = g(x) + c. Using differentiation rules, you can take the derivative of each term separately. Known as the Sum Rule in differentiation, this says that the derivative of a sum is the sum of the derivatives. For the term g(x), which is a function of x, you would compute its derivative normally to find g'(x). For the constant term c, as we've learned, the derivative is zero. Therefore, the derivative of f(x) is simply g'(x) + 0, which simplifies to just g'(x).

This example underscores the importance of being familiar with the differentiation rules in order to correctly compute derivatives of more complex functions.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a cornerstone concept that links the process of differentiation with that of integration. It is one of the most powerful tools in calculus and comprises two parts that together form a deep connection between these two core operations.

The first part of the theorem essentially states that the integral of a function over an interval can be found using the antiderivative of that function. Mathematically, if F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b is F(b) - F(a).

The second part tells us that the derivative of the integral of a function is the original function itself. This means if you take a function f(x), integrate it, and then differentiate this integral, you end up with the original function f(x).

Together, these concepts mean that differentiation and integration are inverse processes. This fundamental relationship allows for a deeper understanding of calculus and provides methods for solving complex problems in mathematics and applied fields.