# 10.3 Differentiating polynomials

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)