Chapter 13 - Section 13.3 - Arc Length and Curvature - 13.3 Exercise - Page 868: 27

$\dfrac{12x^2}{(1+16x^6)^{3/2}}$

Given: $y=x^4$ Consider $f(x)=y=x^4$ In order to find the curvature we will have to use formula 11, such that $\kappa(x)=\dfrac{|f''(x)|}{[1+(f'(x))^2]^{3/2}}$ $y'=4x^3$ and $y''=12x^2$ $|y''|=\sqrt{(12x^2)^2}=12x^2$ $\kappa(x)=\dfrac{|12x^2|}{[1+(4x^3))^2]^{3/2}}$ Hence, $\kappa(x)=\dfrac{12x^2}{(1+16x^6)^{3/2}}$