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The Power Rule

April 9th, 2009 |

Author: Merrin

The power rule is for the derivative of a power function with respect to x which is defined as

$f(x) = x^n \qquad -\infty<x<\infty \qquad n\in {\mathbb R}$

In this section, we will show the power rule for positive integer powers.

$\cfrac{d}{dx} x^n = nx^{n=1}$

The fact that it is true for all real exponents we will just have to quote as a theorem for now. We have already derived a special case when n equals zero. The derivative of a constant function is zero. For exponents less than one be careful that the derivative at x=0 is undefined. Division by zero is not defined.

As we learn more differentiation rules in a certain order it will be possible to show the power rule for more and more cases. First positive integers, then negative integers, then rational powers, and then all real numbers.

Example 1. Apply the definition of the derivative to reveal the power rule for the first few positive integers.

Solution 1. Differentiation of the first few powers of the nth power of x for n = 1,2,3 are sufficient to reveal the power rule.

The derivative of the first power of x is one.

${\cfrac{d}{dx} x}={\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{x+\Delta x-x}{\Delta x} =\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{\Delta x}{\Delta x} =1}$

The derivative of the second power of x is 2x.

${\cfrac{d}{dx} x^{2} }{=}{\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{(x+\Delta x)^{2} -x^{2} }{\Delta x} =\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{x^{2} +2x\Delta x+(\Delta x)^{2} -x^{2} }{\Delta x} }$

$[x^2]'={\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{2x\Delta x+(\Delta x)^{2} }{\Delta x} =\mathop{\lim }\limits_{\Delta x\to 0} \left( 2x+\Delta x \right)=2x}$

The derivative of the third power of x is 3 x squared.

${\cfrac{d}{dx} x^{3} }={\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{(x+\Delta x)^{3} -x^{2} }{\Delta x} =\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{x^{3} +3x^{2} \Delta x+3x(\Delta x)^{2} +(\Delta x)^{3} -x^{3} }{\Delta x} }$

$[x^3]'={\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{3x^{2} \Delta x+O[(\Delta x)^{2} ]}{\Delta x} =\mathop{\lim }\limits_{\Delta x\to 0} 3x^{2} +O[\Delta x]=3x^{2} }$

From the pattern we can jump to positive integer n.

$\begin{array}{ll}(x+\Delta x)^{1}&\!\!=x+\Delta x \\(x+\Delta x)^{2} &\!\!=x^{2} +2x\Delta x+(\Delta x)^{2}\\(x+\Delta x)^{3} &\!\!=x^{3} +3x^{2} \Delta x+3x(\Delta x)^{2} +(\Delta x)^{3}\\(x+\Delta x)^{n} &\!\!=x^{n} +nx^{n-1} \Delta x+O[(\Delta x)^{2} ]\end{array}$

$(x+\Delta x)^{n} = \sum_{k=0}^{n}\cfrac{n!}{k!(n-k)!} x^{n-k} \Delta x^{k}$

Only the first two terms from the binomial theorem contribute to the derivative of x to the n, the later terms are proportional to higher orders of the differential and their part in the limit ends up being zero. The second term is always

$nx^{n-1} \Delta x$

The differential cancels out in the limit and we have the power rule for positive integers.

$\cfrac{d}{dx} x^{n} = \mathop{\lim }\limits_{\Delta x\to 0} \cfrac{(x+\Delta x)^{n} -x^{n}}{\Delta x} = \mathop{\lim }\limits_{\Delta x\to 0} \cfrac{(x)^n +nx^{n-1} -x^n + O[(\Delta x)^2]}{\Delta x}$

$f'(x)= \mathop{\lim }\limits_{\Delta x\to 0}\cfrac{nx^{n-1}\Delta x}{\Delta x} = nx^{n-1}$

Example 2. Find the derivative of

$f(x)=1+x^{2} +x^{3}$

Solution 2. The derivative can be broken up into a sum from the basic properties of derivatives.

$\cfrac{d}{dx} (1+x^{2}+x^{3} )=\cfrac{d}{dx} 1+\cfrac{d}{dx} x^{2}+\cfrac{d}{dx} x^{3} =0+2x+3x^{2}$

Example 3. Find the derivative of

$f(x)=9x+7x^{2} +20x^{9}$

Solution 3. For this problem we use the rule for differentiation functions multiplied by constants, and then differentiate term by term.

$\cfrac{d}{dx} f(x)=\cfrac{d}{dx} 9x+\cfrac{d}{dx} 7x^{2} +\cfrac{d}{dx} 20x^{9}$

$f'(x)=9\cfrac{d}{dx} x+7\cfrac{d}{dx} x^{2} +20\cfrac{d}{dx} x^{9}$

$f'(x)=9\cdot 1x^{(1-1)} +7\cdot 2x^{2-1} +20\cdot 9x^{9-1}$

$f'(x)=9+14x+180x^{8}$

Example 4. Find the derivative of

$f(x)=1+x+\cfrac{1}{2!} x^{2} +\cfrac{1}{3!} x^{3} +...+\cfrac{1}{n!} x^{n} +...$

Solution 4. We can use the power rule on each term

${\cfrac{d}{dx} f(x)}{=} {\cfrac{d}{dx} 1+\cfrac{d}{dx} x+\cfrac{d}{dx} \cfrac{1}{2!} x^{2} +\cfrac{d}{dx} \cfrac{1}{3!} x^{3} +...+\cfrac{d}{dx} \cfrac{1}{n!} x^{n} +...}$

$f'(x)= {0+1+\cfrac{1}{2!} 2x+\cfrac{1}{3!} 3x^{2} +...+\cfrac{n}{n!} x^{n-1} +...}$

$f'(x){=}{1+x+\cfrac{1}{2!} x^{2}+...+\cfrac{1}{(n-1)!} x^{n-1} +...}={f(x)}$

That function turns out to be its own derivative!

Example 5. The most general polynomial can be written as

$f(x)=a_0+a_1x+a_2x^2 + \cdots$

Find the derivative of f(x).

Solution 5. Applying the power rule to each term we find the derivative is just

$f'(x)=a_1+2*a_2x+3a_3x^2+\cdots$

Example 6. Sometimes complex expressions can be put in a form that is easily

solvable by the power rule. Find the derivative of

$f(x) = \sqrt[{4}]{x^{3} } +\cfrac{1}{\sqrt{x} } +x^{2/3} \left(\cfrac{1}{x^{9/5}} + x^{2} \right)$

Solution 6. Write all the powers as fractions in the numerator, “upstairs.”

$f(x)=\sqrt[{4}]{x^{3}} +\cfrac{1}{\sqrt{x}} + x^{2/3} \left(\cfrac{1}{x^{9/5}} + x^{2} \right)=x^{3/4} +x^{-1/2} +x^{-17/15} +x^{8/3}$

${\cfrac{d}{dx} f(x)=\cfrac{3}{4} x^{-1/4} -\cfrac{1}{2} x^{-3/2} -\cfrac{17}{15} x^{-32/15} +\cfrac{8}{3} x^{5/3}}$

Don’t be intimidated by simple expressions that look complex. The power rule only applies to power functions. Don’t get creative and try to fit other functions into the mould. The power rule does not apply to

$f(x)=a^x \qquad \text{or} \qquad f(x) = x^x$

We will learn how to differentiate this function in turn, but for now just look for a variable in the base and a number in the exponent. Later we will learn how to differentiate

$[f(x)]^n$