SOLUTION 11: Begin with the function
$$f(x)= \arctan x$$
and choose
$$x-values: 1 \rightarrow 1.1$$
so that
$$\Delta x = 1.1-1 = 0.1$$
The derivative of $\ y=f(x) \$ is
$$f'(x)= \displaystyle{ 1 \over 1+x^2 }$$
The exact change of $y-$values is
$$\Delta y = f(1.1) - f(1)$$ $$= \arctan(1.1) - \arctan(1)$$ $$= \arctan(1.1) - \displaystyle{ \pi \over 4 }$$ $$\approx \arctan(1.1) - 0.7854$$
The Differential is
$$dy = f'(1) \ \Delta x$$ $$= \displaystyle{ 1 \over 1+(1)^2 } \cdot (0.1)$$ $$= \displaystyle{ 1 \over 2 } (0.1)$$ $$= 0.05$$
We will assume that
$$\Delta y \approx dy \ \ \ \ \longrightarrow$$ $$\arctan(1.1) - \displaystyle{ \pi \over 4 } \approx 0.05 \ \ \ \ \longrightarrow$$ $$\arctan(1.1) \approx \displaystyle{ \pi \over 4 } + 0.05 \ \ \ \ \longrightarrow$$ $$\arctan(1.1) \approx 0.7854 + 0.05 \ \ \ \ \longrightarrow$$ $$\arctan(1.1) \approx 0.8354$$

NOTE: The number 1 was chosen for its proximity to 1.1 and for it's convenient inverse tangent. Check the accuracy of the final estimate using a CALCULATOR: $\arctan(1.1) \approx 0.8330$