Discussion Forum

If f(x) =f(t)  and  y= g(t) are differentiable functions of t, then $\frac{d^{2}y}{dx^{2}}$  is 

Answer:

Explanation:

 Given , x=f(t) and y=g(t)

 On digfferentiating  both sides w.r.t 't' , we get

$\frac{dx}{dt}=f'(t)$and  $\frac{dy}{dt}=g'(t)$

 We know that,  $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

 $\Rightarrow$    $\frac{dy}{dx}=\frac{g'(t)}{f'(t)}$

 Again, differentiating both sides w.r.t 'x' , we get

 $\frac{d^{2}y}{dx^{2}}=\frac{f'(t).g''(t)-g'(t).f''(t)}{(f'(t))^{2}}.\frac{dt}{dx}$

    =$\frac{f'(t).g''(t)-g'(t).f''(t)}{(f'(t))^{2}}.\frac{1}{f'(t)}$

                               $=\frac{f'(t).g''(t)-g'(t).f''(t)}{(f'(t))^{3}}$