Chapter 13 - Vector Functions - 13.3 Arc Length and Curvature - 13.3 Exercises - Page 908: 27

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

As we are given that $y=x^4$ Let $f(x)=y=x^4$ ...(a) Write formula 11. $\kappa(x)=\dfrac{|f''(x)|}{[1+(f'(x))^2]^{3/2}}$ $f'(x)=4x^3$ and $f''(x)=12x^2$ [ from equation (a)] or $|f''(x)|=\sqrt{(12x^2)^2}=12x^2$ Thus, $\kappa(x)=\dfrac{|12x^2|}{[1+(4x^3))^2]^{3/2}}$ or, $\kappa(x)=\dfrac{12x^2}{(1+16x^6)^{3/2}}$