Looking for tutors that are experts in derivatives.
Hello tutors. Can you give 3 example functions where the constant rule (derivatives) is applicable? Examples with detailed answers and solutions. I am having a hard time finding example functions with solutions on the internet and I wanted to review the concept of it. (Not constant multiple, just constant)
The correct answer and explanation is :
Certainly! The constant rule for derivatives is a fundamental concept in calculus. The constant rule states that the derivative of any constant (a number with no variable) is always 0. This rule is helpful for simplifying the differentiation process when we encounter constants in mathematical functions. Let’s go through three example functions where the constant rule applies.
Example 1:
Function: f(x)=5f(x) = 5
Solution: The derivative of a constant function is always 0. Since f(x)=5f(x) = 5 is a constant function (the value doesn’t change regardless of the value of xx), we apply the constant rule: f′(x)=0f'(x) = 0
Explanation: The function f(x)=5f(x) = 5 represents a horizontal line on the graph, where the slope of the line is 0 everywhere. Since the slope of a horizontal line is always 0, the derivative is also 0.
Example 2:
Function: g(x)=−3g(x) = -3
Solution: Again, g(x)=−3g(x) = -3 is a constant function. Applying the constant rule: g′(x)=0g'(x) = 0
Explanation: Just like the previous example, the function g(x)=−3g(x) = -3 represents a horizontal line, but this time it is below the x-axis. The slope is still 0, so the derivative is 0.
Example 3:
Function: h(x)=12h(x) = 12
Solution: For the function h(x)=12h(x) = 12, which is a constant, the derivative is: h′(x)=0h'(x) = 0
Explanation: This is yet another example of a constant function. The function is a horizontal line at y=12y = 12, and the slope of a horizontal line is 0. Therefore, the derivative of the function is 0.
Summary of the Constant Rule:
The constant rule states that the derivative of any constant function is always 0. This rule is intuitive because a constant function does not change with respect to xx, so there is no rate of change, and thus, the derivative is 0.
By recognizing and applying this rule, you can simplify the process of differentiating more complex functions where constants appear, and understand how they behave in the context of calculus.