Which of the following functions is differentiable at x = 0?

Linear combination of one differentiable and one non-differentiable function is always nondifferentiable.

Hence (a) and (b) are wrong.

Now, $\tan(|x|)-|x|=\frac{|x|^3}{3}+\frac{2|x|^5}{15}+.....$ is differentiable at x = 0,

whereas $\tan(|x|)+|x|=2|x|+\frac{|x|^3}{3}+\frac{2|x|^5}{15}+.....$ is not differentiable at x = 0.

[∵ |x| is not diff. at x = 0]