Question

If, y = tan^-1(x/a), then what is the value of y’?

(a) (2ax/(x^2 – a^2)^2)

(b) -(2ax/(x^2 – a^2)^2)

(c) (2ax/(x^2 + a^2)^2)

(d) -(2ax/(x^2 + a^2)^2)

I got this question in unit test.

Origin of the question is Second Order Derivative topic in division Limits and Derivatives of Mathematics – Class 11

Answer 1

Correct choice is (d) -(2ax/(x^2 + a^2)^2)

Easy explanation: We have, y = tan^-1(x/a)

Differentiating two times, with respect to x, we get,

y’ = d/dx(tan^-1(x/a))

= 1/(1 + (x/a)^2)*d/dx(x/a)

= a^2/(x^2 + a^2)* 1/a

= a/(x^2 + a^2)

y” = d/dx (a/(x^2 + a^2))

= a* d/dx(x^2 + a^2)^-1

= a(-1) (x^2 + a^2)^-2*d/dx(x^2 + a^2)

Now solving we get,

= -a/(x^2 + a^2)^2*2x

= -(2ax/(x^2 + a^2)^2)