Question

If y = tan-1 x, show that ( 1 + x2) d2y / dx2 + 2x dy/dx = 0

Answer 1

Given : y = Tan⁻¹x

To Prove :  (1 + x²) (d²y/dx²)  + 2x(dy/dx)

Solution:

y = Tan⁻¹x

Taking Tan both sides

=> Tan y = x

Differentiating both side wrt x

=> Sec²y dy/dx  = 1

Using Sec²y = 1 + Tan²y

=> (1  + Tan²y)dy/dx  = 1

Tany = x => Tan²y = x²

=> (1 + x²)dy/dx  = 1

Differentiating again wrt x

=>  (1 + x²) (d²y/dx²)  + (0 + 2x)dy/dx  = 0

=>  (1 + x²) (d²y/dx²)  + 2x(dy/dx)  = 0

QED

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Answer 2

To show ( 1 + x2) d2y / dx2 + 2x dy/dx = 0

• Given, y = tan^(-1) x

• Differentiating y we get, dy/dx = 1 / (1 + x^2)

• From rearranging above equation we get, (1 + x^2) * dy/dx = 1

• Differentiating again, using the rule of differentiating product of two: {Rule: d(uv)/dx = u* dv/dx + v* du/dx}

                 (1 + x^2) * (d2y/d2x) + 2x (dy/dx) = d(1)/dx

                 (1 + x^2) * (d2y/d2x) + 2x (dy/dx) = 0