# Differentiating Polynomials

It can be shown by differentiating from first principles that

$\frac{d}{dx}\left(x^n\right)=nx^{n-1}$

For example, if $y=x^3$ then $\frac{dy}{dx}=3x^2$ and it follows that the point (2,8) on the cubic graph has a gradient of 12.

The principle can be applied to linear combinations of powers of $x$, also known as polynomials. For example, the polynomial $f(x)=5x^2-2x^4$ has the derivative $f$. One can think of it informally as ‘multiplying down by the power, then taking one off of the power’.

Note that, in maths, differentiation is finding the derivative or gradient function. The gradient of the straight line, $y=4$ for example, is zero and so the derivative of a constant is 0.

Example 1:
$y=4x^{2}-5x^{-1}+7, \frac{dy}{dx}=8x+5x^{-2}$

Example 2:
$f(x)=8x^{-\frac{1}{2}}+3x^{\frac{1}{2}}, f$