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Taking the Derivative of 5x^2: How-To & Steps

Instructor: Laura Pennington

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

In this lesson, we will find the derivative of 5x^2. We will learn a general formula we can use to find this derivative and we will also look at how to use the limit definition of the derivative to find this derivative as well.

Steps to Solve

We are going to learn how to find the derivative of 5x2. This may seem a bit intimidating at first glance, but this is one of those derivatives that we have a nice formula for, and that makes our job a whole lot easier! Notice that 5x2 is in the form axn, where a is a constant. Well, like we just said, we have a general formula for the derivative of axn that we can apply to 5x2, and that is as follows:

If f(x) = axn, then f ' (x) = nax(n-1)

That's not so bad! All we have to do is identify our a and our n, and then plug those values into the formula, and we have our derivative!

In the case of 5x2, we have that a = 5 and n = 2. Therefore, we plug these into the formula and simplify.

nax(n-1) = 2⋅5x(2-1) = 10x1 = 10x

We see that the derivative of 5x2 is 10x. Wow! That formula sure makes things easy!

Solution

The derivative of 5x2 is 10x.


derv5x21


Using the Limit Definition of a Derivative

We just saw how to find the derivative of 5x2 using the general formula for the derivative of the function axn. But, what happens if you forget the formula? Don't worry, all hope is not lost! You see, we can use the limit definition of the derivative of a function to find this derivative as well. This process is a bit more involved, but let's just take it one step at a time and see how it turns out.

First of all, let's state the limit definition of a derivative.


derv5x22


We see that the derivative of f(x) can be found by finding the limit, as h→0, of (f(x + h) - f(x)) / h. Okay, easy enough! Let's give this a go with our function f(x) = 5x2!

First, lets find f(x + h), so we can plug that into the limit and simplify.

f(x + h) = 5(x + h)2 = 5(x + h)(x + h) = 5(x2 + 2xh + h2) = 5x2 + 10xh + 5h2

Alright, now we plug f(x + h) = 5x2 + 10xh + 5h2 and f(x) = 5x2 into the limit definition of the derivative and simplify.


derv5x23


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