10.3 Differentiating polynomials ‣ Chapter 10 Differential calculus ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd‣ Chapter 10 Differential calculus ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd

To differentiate $f(x)=x^{n}$ some algebra is required⁷⁷ 7 If you’re not confident in conducting algebraic manipulations (i.e. \saydoing algebra) then it really is worth spending time reviewing this; it’s foundational for everything else in mathematics.

$\displaystyle\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$	$\displaystyle=\lim_{h\to 0}\frac{(x+h)^{n}-x^{n}}{h}$	(10.17)
	$\displaystyle=\lim_{h\to 0}\frac{x^{n}+\binom{n}{1}x^{n-1}h+\binom{n}{2}x^{n-2% }h^{2}+...+h^{n}-x^{n}}{h}$	(10.18)
	$\displaystyle=\lim_{h\to 0}\frac{\binom{n}{1}x^{n-1}h+\binom{n}{2}x^{n-2}h^{2}% +...+h^{n}}{h}$	(10.19)
	$\displaystyle=\lim_{h\to 0}\binom{n}{1}x^{n-1}+\binom{n}{2}x^{n-2}h+...+h^{n-1}$	(10.20)
	$\displaystyle=nx^{n-1}$	(10.21)

Note that in the process of carrying out the expansion

•

In Equation 10.18 we used the binomial theorem (as in Equation 3.64).
•

In Equation 10.19 we used the fact that $x^{n}+(-x^{n})=0$
•

In Equation 10.20 we divided through by $h$ .
•

In the final step, we applied the property that $h\cdot X$ (where $X$ is some expression⁸⁸ 8 Where $X\in\mathbb{R}$ ) is $0$ as $h\to 0$ .

We can then combine this with the rule for the derivatives of sums from above to find the derivatives of any polynomial.

For example, we can find the derivative of $x^{2}+3x-8$ (which was the example used above).

	$\displaystyle\frac{d}{dx}(x^{2}+3x-8)$	$\displaystyle=\frac{d}{dx}[x^{2}]+\frac{d}{dx}[3x]+\frac{d}{dx}[-8]$		(10.22)
		$\displaystyle=2x+3$		(10.23)

Why is $\frac{d}{dx}(-8)=0$ ?

Let’s suppose we have a function $f(x)=c$ , then the derivative of $f(x)$ is just

	$\displaystyle\lim_{h\to 0}\frac{-8-(-8)}{h}$	$\displaystyle=\lim_{h\to 0}\frac{0}{h}$		(10.24)
		$\displaystyle=0$		(10.25)