Description
Given 3x = y^2 - x^2 , find d^2y/dx^2
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Explanation & Answer
3x = y^2 - x^2
Now
y^2 = x^2 + 3x
Taking derivative w.r.t x, then
d/dx(y^2) = d/dx(x^2 + 3x)
2y dy/dx = 2x + 3
Now
dy/dx = (2x + 3)/2y
As
y = √ (x^2 + 3x) so
dy/dx = (2x + 3)/(√(x^2 + 3x))
Again taking derivative w.r.t x, we get
d/dx(dy/dx) = d/dx[ (2x + 3) / √(x^2 + 3x) ]
Using the rule of d/dx(u/v) = [v*d/dx(u) - u*d/dx(v)] / v^2
d^2 y/dx^2 = [ √(x^2 + 3x) * d/dx(2x + 3) - (2x + 3) * d/dx(√(x^2 + 3x)) ] / (√x^2 + 3x)^2
= [ 2*(√x^2 + 3x) - (2x + 3)(1/2 *√(x^2 + 3x) * (2x + 3)) ] / (x^2 + 3x)
= [ 2√(x^2 + 3x) - (2x + 3)^2 / 2√(x^2 + 3x) ] / (x^2 + 3x)
= [ {4(x^2 + 3x) - (2x + 3)^2} / 2√(x^2 + 3x) ] / (x^2 + 3x)
= (4x^2 + 12x - 4x^2 - 9 - 12x) / (x^2 + 3x) * 2√(x^2 + 3x)
= - 9 / 2(x^2 + 3x)^3/2
Hence d^2 y/dx^2 = - 9 / 2(x^2 + 3x)^3/2
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