Question

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Constant. $$y=3 b+7 c$$

Short answer

The derivative of the function with respect to a variable, which is absent in the function, is zero. Therefore, the derivative of \(y = 3b + 7c\) with respect to any variable is zero.

Step-by-step solution

Identify Function Type

Recognize that the function consists of constants multiplied by variables.

Apply Constant Rule

Remember that the derivative of a constant multiplied by a variable with respect to that variable is just the constant. The derivative of a constant alone is zero.

Calculate Derivatives

Apply the constant rule to both terms in the function separately.

Combine the Results

Since the derivative of a constant is zero, terms involving only constants disappear.

Key concepts

Derivative of a Constant

The derivative of a constant is one of the fundamental principles in calculus that frequently perplexes students. To demystify this, imagine any constant value; let's use the number 5 as an example. If you were to draw a graph of the function \(y = 5\), it would be a horizontal line. No matter how far along the x-axis you go, \(y\) would always be 5. Now, derivatives represent the rate of change, and a horizontal line does not change as it moves along the x-axis; it remains constant. Consequently, the derivative of any constant such as \(y = 5\) is simply \(0\).

When you're working with a function that includes constants, such as \(y = 3b + 7c\), you treat each constant term separately. If \(b\) and \(c\) are constants, then so is \(3b\) and \(7c\), and their derivatives will also be zero when taken with respect to a variable that does not appear in the term.

Constant Rule in Calculus

The constant rule in calculus is a straightforward yet invaluable tool for computing derivatives. It states that the derivative of any constant multiplied by a variable is simply the constant itself. To understand why, consider a function like \(y = ax\), where \(a\) is a constant and \(x\) is the variable. The graph of \(y = ax\) would be a straight line that crosses the origin, and its slope at any given point is \(a\). Since the derivative is equivalent to the slope of the function at any point, the derivative of \(y = ax\) with respect to \(x\) is \(a\).

Using this rule simplifies calculating derivatives, as you can see in the textbook exercise where \(y = 3b + 7c\). Applying the constant rule, you realize that since \(b\) and \(c\) are not variables, their coefficients (3 and 7) are the only focus, and the derivative of each term with respect to \(x\) would be \(0\) because they do not depend on \(x\) at all.

Calculus Step by Step Solution

Working through calculus problems step by step is essential to understanding and mastering the processes involved. Calculus can seem daunting, but breaking down the process into smaller stages can significantly simplify the task. Let's use our textbook exercise as a practical example on how to apply a step by step solution.

First, identify the type of function you are dealing with. In this case, \(y = 3b + 7c\) contains constants and no variables. Next, apply the mathematical rules that are relevant to the problem, such as the constant rule we've discussed. From there, calculate the derivatives of each term in the function. Since both \(3b\) and \(7c\) are constants, their derivatives are zero. Finally, combine these results to construct the complete derivative. The meticulous nature of this approach ensures that you understand each part of the problem and the reason behind every step.