Deriving the Derivative of a Monomial

We know that if k is a constant then

$\dfrac {d}{dx} k f(x)= k \dfrac {d} {dx} f(x)$

We can use this after we do the work to find the derivative of $x^n$ .

We are going to do the work for a scenario where n is an integer.

The equation below shows the limit that calculates the derivative of a function.

$\displaystyle lim_{t \to 0} \dfrac {f(x+t) - f(x)} {(x+t) - x}$

$\displaystyle lim_{t \to 0} \dfrac {f(x+t) - f(x)} {t}$

Let’s work a few examples and see if we notice anything that could be helpful…

Let n=1:

$f(x)=x^1=x$

$f(x+t) = x+t$

$\displaystyle lim_{t \to 0} \dfrac {f(x+t) - f(x)} {t}$

$\displaystyle = lim_{t \to 0} \dfrac {x+t x} {t}$

$\displaystyle = lim_{t \to 0} \dfrac {t} {t}$

$\displaystyle = lim_{t \to 0} 1$

$= 1$

Let n=2:

$f(x)=x^2$

$f(x+t)=x^2+2xt+t^2$

$\displaystyle \lim_{t \to 0} \dfrac {x^2+2xt +t^2 - x^2}{t}$

$\displaystyle \lim_{t \to 0} 2x +t = 2x$

As the integer values for n get higher and higher, the number of terms from the first term in the limit definition, the f(x+t), increases. Our goal is to try and notice a trend after working several examples. Continuing with the math (see Appendix A for the calculation of $(x+t)^n$ , we get the following:

for n=3:

$\displaystyle \lim_{t \to 0} \dfrac {x^3 + 3x^2t + 3xt^2 + t^3 - x^3}{t}$

$\displaystyle \lim_{t \to 0} 3x^2 + 3xt + t^2$

$3x^2$

for n=4:

$\displaystyle \lim_{t \to 0} \dfrac {x^4 + 4x^3t + 6x^2t^2 + 4xt^2 + t^4 - x^4}{t}$

$\displaystyle \lim_{t \to 0} 4x^3 + 6x^2t + 4xt + t^3$

$4x^3$

We see a trend of the following:

The first term of the expansion of $(x+t)^n$ cancels with the second term of the numerator
The division of t removes the t from the second term and this term survives
The third term in the expansion, and all terms after it, have t to the power of 1 or higher and so the limit causes them to go to zero, thus they do not survive

The answer for n is always $nx^{n-1}[latex] Appendix A: To take n to values higher and higher, we need to do quite a bit of multiplication of of the factor "x+t": <figure class="wp-block-image size-large"><img data-attachment-id="2089" data-permalink="https://rob-sterling.com/induction/binomialmultiplication/" data-orig-file="https://rob-sterling.com/wp-content/uploads/2020/06/binomialmultiplication.jpg" data-orig-size="508,620" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"1"}" data-image-title="binomialMultiplication" data-image-description="" data-image-caption="" data-medium-file="https://rob-sterling.com/wp-content/uploads/2020/06/binomialmultiplication.jpg?w=246" data-large-file="https://rob-sterling.com/wp-content/uploads/2020/06/binomialmultiplication.jpg?w=508" width="508" height="620" src="https://rob-sterling.com/wp-content/uploads/2020/06/binomialmultiplication.jpg?w=508" alt="" class="wp-image-2089" srcset="https://rob-sterling.com/wp-content/uploads/2020/06/binomialmultiplication.jpg 508w, https://rob-sterling.com/wp-content/uploads/2020/06/binomialmultiplication.jpg?w=123 123w, https://rob-sterling.com/wp-content/uploads/2020/06/binomialmultiplication.jpg?w=246 246w" sizes="(max-width: 508px) 100vw, 508px" /></figure> Fortunately after doing a few multiplications we notice a trend. There are constants which follow a pattern that someone noticed to be "<a href="https://rob-sterling.com/pascals-triangle/">Pascal's Triangle</a>" and for n multiplications of "x+t" the answer fits the following form (not showing the constants): [latex]...x^nt^0 + ...x^{n-1}t^1 + ...x^{n-2}t^2 + ...x^2t^{n-2} + ...x^1t^{n-1} + ...x^0t^n$

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