Hippocampus Calculus: Constant rule, constant multiple rule

The process of finding the derivative of a function is known as differentiation.

Although we can differentiate a function by using our definition of a derivative as a limit, doing so is generally tedious. The differentiation process can be simplified by learning some useful rules, which we develop in this section.

Suppose we have a function f(x) that is equal to a constant. The graph of this function is a horizontal line.

Therefore, the slope at any point P along this line is zero, because there is no change of y over any change of x.

It then follows that the derivative at any point P must also be zero.

The foregoing illustrates the constant rule, which states that the derivative of a constant function is zero.

We can prove the constant rule by using our definition of the derivative.

After substituting the constant for the function in the limit equation, the evaluation of the limit is trivial.

The derivative is equal to zero, as the rule states.

Suppose we have a function f(x), and a constant c = 2. We can create a new function g(x) by multiplying f(x) by the constant c.

The graph of the two functions shows that the curve g(x) is displaced vertically from f(x) by a factor of two.

How do you suppose lines tangent to the two curves at the same value of x differ? The slope of the tangent line to g(x) is twice that of the tangent line to f(x).

This illustrates the constant multiple rule, which states that the derivative of a constant times a function is equivalent to the constant times the derivative of the function.

Let's look at our example once again. According to the constant multiple rule, the derivative of g(x) should be twice the derivative of f(x) for any value of x where the derivative exists.

We can use graphical means to confirm that this is the case. The graph shown displays the derivatives of our two functions at a selected value of x; we know that the derivatives are just the slopes of the lines tangent to the curves at each value of x.

Use the slider to vary the value of x. Note the relationship between the derivatives.

Next let's verify the constant multiple rule. We begin with the definition of the derivative.

After substituting our function into the equation, we factor out the constant c from the main limit term.

Then, to bring the constant c outside the limit, we use the limit law, which states that the limit of a constant times a function is the constant times the limit of the function.

We see that the remaining limit term is just f′(x). The result is just the constant multiple rule.

Now, use the constant multiple rule to determine which of the following is the derivative of the given function g(x). Click the "Submit" button after selecting your answer.