Science:Math Exam Resources/Courses/MATH100/December 2018/Question 01 (c)/Solution 1

Let $f(x)=\arctan(x^{2})$, then the tangent line at $x=1$ will be given by the equation $y-f(1)=f'(1)(x-1)$.

Using the chain rule, we get $f'(x)={\frac {1}{1+(x^{2})^{2}}}\cdot 2x={\frac {2x}{1+x^{4}}}$, and therefore $f'(1)=1$. Plugging this and $f(1)={\frac {\pi }{4}}$ into the equation for the tangent line gives $y-{\frac {\pi }{4}}=x-1$. After rearranging, we conclude that the tangent line is $\color {blue}y=x+{\frac {\pi }{4}}-1$.