Solving the Derivative of 2x

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will show the formula we can use to find the derivative of 2x. It will also show another way of finding this derivative using different properties and rules. We'll finish off with an application of this specific problem that could show up in our daily lives.

Steps to Solve

To find the derivative of 2x, we can use a well-known formula to make it a very simple process. The formula for the derivative of cx, where c is a constant, is given in the following image.


der2x8


Since the derivative of cx is c, it follows that the derivative of 2x is 2.

Solution

We've used the formula to find that the derivative of 2x is 2.

Another Solving Process and an Application

Though the formula for the derivative of cx, where c is a constant, is well-known, one could find themselves in a situation when they don't know it or they can't recall it. Therefore, it is useful to know another way of solving the problem. Let's look at a way to do this.

This method of finding the derivative is longer and you need to be familiar with a few rules and properties to be able to carry it out. It's still a great way of finding the derivative though, especially if you're not familiar with the formula! Let's examine the properties and rules we'll need to know to carry out this process.

  • The product rule of derivatives states the following.


der2x5


  • The derivative of x a is ax a-1
  • The derivative of a constant is 0.
  • x 0 = 1

Alright, let's use these facts to find the derivative of 2x.

Notice that 2x is a product of the functions f(x) = 2 and g(x) = x. This tells us we can use the product rule to find the derivative. We plug f(x) = 2 and g(x) = x into the product rule as follows.


der2x2


Now we simplify by finding the derivative of 2 and of x. By the facts, the derivative of 2 is 0. To find the derivative of x, we can think of it as x 1 and use our fact. Thus, the derivative of x is 1*x 1-1 = x 0 = 1. Let's plug these in and simplify.


der2x3


We see that we get that the derivative for 2x is 2, same as when we used our formula.

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