Derivative Power Rule

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Introduction

So far we have calculated derivatives using the definition of the derivative and manually evaluating the limit:

(1)   \begin{equation*} f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \end{equation*}

It’s likely that you’ve looked at basic examples with this formula, including small power functions or polynomials, and perhaps a few other types of functions. However, in practice, this method is tedious and becomes incredibly difficult for more advanced functions. Ideally, we’d like to use this definition in order to glean more information about the derivative so we can generalize the process for functions without having to revisit this challenging formula every time.

We will begin our (ultimately very long) exploration of the derivative (which is most of the rest of this class) by finding a general rule for basic power functions of the form f(x) = x^n, where ultimately n is any number. It turns out thatwe can use the definition of the derivative to produce a rule for differentiating such functions in a general way; this prevents us from having to use the definition each and every time a power function is differentiated.

After a lot of plugging in and simplifying, we find the power rule of differentiation:

Check out my full notes on this section for an extensive proof of this result using the definition of the derivative, as well as a proof using later calculus and numerous other practice problems and explanation!

As always, I prefer a word-based memorization of such formulas instead of a mathematical one. Say “the derivative of x to some power is that power times x raised to the power of the old exponent minus one”.

You are now at a point where using the definition of the derivative (except on test or homework problems that specifically ask for it) will no longer be necessary. All of your calculus work from here on out will revolve around finding the condensed, simplified formulas for various types of derivatives and then applying them to various problems. Taking the derivative of x^5 will now be as simple as writing 5x^4 instead of using the definition.

Examples

Let’s check out some examples. I’ve made a deliberate choice in the organization of these notes to only look at examples whose functions directly line up with the form of the power rule above, such as x^5, x^{-13} or \sqrt{x} (can you see how this one fits into that form?) Other more advanced examples, such as 32x^5, x^{-13} + x^2 - x^4, or \sqrt{x} -2, will be discussed in the next section on properties of the derivative!

Example 1

Let f(x) = x^{7}. Find f'(x).

Computing such derivatives is now as simple as utilizing the power rule above. The power of this function, n, is 7. Therefore, the derivative is simply

    \[ f'(x) = 7x^6 \]

Example 2

Find the derivative of h(t) = t^{100}.

It dosen’t matter how large the power is; we can proceed as normal:

    \[ h'(t) = 100t^{99} \]

Example 3

Compute the derivative of y(x) = x^{-6}

Negative values are fair game too! The power rule produces

    \[ y'(x) = -6x^{-7} \]

Example 4

Find the derivative of z = \sqrt{x}

This derivative can also be taken using the power rule. Try it out and see if you can find how before proceeding.

Recall that the square root function can be written as an exponent of 1/2: \sqrt{x} = x^{1/2}. Study up on your algebra and exponent rules and proceed with the power rule:

    \[ \diff{}{x} \Big[x^{1/2} \Big] = \frac{1}{2} x^{(1/2) - 1} = \frac{1}{2}x^{-1/2} \]

Recall that negative powers imply reciprocals:

    \[ = \frac{1}{2} \; \frac{1}{x^{1/2}} = \frac{1}{2\sqrt{x}} \]

There will be a lot of this sort of algebraic manipulation for these problems.

Example 5

Find the derivative of y = \sqrt[4]{x^5}.

Rewriting this function as y = x^{5/4}, we can proceed with the power rule.

    \[ y' = \frac{5}{4} x^{1/4} = \frac{5}{4} \sqrt[4]{x} \]

Example 6

Find the derivative of x = \frac{1}{m^3}.

We can rewrite x = m^{-3}, which now matches the power rule exactly. Note that you can’t just take the derivative of the denominator and place it under the 1; the entire function can only be written as a power function if written as x = m^{-3}.

Finding the derivative produces

    \[ \diff{x}{m} = -3m^{-4} = - \frac{3}{m^4} \]

The power rule reduces the power, which increases its absolute value from 3 to 4 when found in the denominator.

Looking for more examples in this section? To start, you may find more useful examples of taking derivatives of more complex functions here. I chose to place these examples in that section because you’ll want to see the properties of the derivative before trying them out.

You’ll also find a few more interesting cases in my full notes on this section, in addition to proofs of the following important results that we will use later on:

    \[ \diff{}{x} \Big[ x \Big] = 1 \]

    \[ \diff{}{x} \Big[ 1 \Big] = 0 \]

The above can be proven with what we’ve learned in this section alone!

Need more? Here you can get the full version of these notes!