Here you will learn higher order derivatives of parametric equations with examples.
Let’s begin –
Higher Order Derivatives of Parametric Equations
We know that the differentiation of parametric equations of type x = at and y = 2at is given by formula
\(dy\over dx\) = \(dy/dt\over dx/dt\)
where t is the parameter
Now, by again differentiating the above equation we obtain 2nd order derivative given below:
\(d^2y\over dx^2\) = \(d\over dx\)(\(dy\over dx\))
Example : find \(d^2y\over dx^2\) , if x = \(at^2\) , y = 2at.
Solution : We have,
x = \(at^2\) , y = 2at.
Differentiating both sides with respect to t,
\(\implies\) \(dx\over dt\) = 2at and \(dy\over dt\) = 2a …….(i)
\(\therefore\) \(dy\over dx\) = \(dy/dt\over dx/dt\) = \(2a\over 2at\) = \(1\over t\)
Differentiating both sides with respect to x, we get
\(d^2y\over dx^2\) = \(d\over dx\)(\(1\over t\))
\(\implies\) \(d^2y\over dx^2\) = \(-1\over t^2\)\(dt\over dx\)
from (i), [\(dx\over dt\) = 2at \(\therefore\) \(dt\over dx\) = \(1\over 2at\)]
\(\implies\) \(d^2y\over dx^2\) = -\(1\over 2at^3\)
Example : If x = \(acos^3\theta\) , y = \(asin^3\theta\) , find \(d^2y\over dx^2\). Also find its value at \(\theta\) = \(\pi\over 6\).
Solution : We have,
x = \(acos^3\theta\) and y = \(asin^3\theta\)
Differentiating both sides with respect to \(\theta\),
\(\therefore\) \(dx\over d\theta\) = \(-3acos^2\theta sin\theta\) and \(dy\over d\theta\) = \(3asin^2\theta cos\theta\) …….(i)
So, \(dy\over dx\) = \(dy/d\theta\over dx/d\theta\) = \(3asin^2\theta cos\theta\over {-3acos^2\theta sin\theta}\) = \(-tan\theta\)
Differentiating both sides with respect to x, we obtain
\(d^2y\over dx^2\) = \(d\over dx\)(\(-tan\theta\))
= \(sec^2\theta\)\(d\theta\over dx\)
from (i), [\(dx\over d\theta\) = \(-3acos^2\theta sin\theta\) \(\therefore\) \(d\theta\over dx\) = \(1\over {-3acos^2\theta sin\theta}\)]
\(d^2y\over dx^2\) = -\(sec^2\theta\)\(\times\)\(1\over {-3acos^2\theta sin\theta}\)
\(\implies\) \(d^2y\over dx^2\) = \(1\over 3a\) \(sec^4\theta\)\(cosec\theta\)
\(\therefore\) At \(\theta\) = \(\pi\over 6\) , \(d^2y\over dx^2\) = \(1\over 3a\) \(sec^4{\pi\over 4}\)\(cosec{\pi\over 6}\)
= \(1\over 3a\) \(\times\) \((2\over \sqrt{3})^4\) \(\times\) 2 = \(32\over 27a\)
