Find second order derivative if- x-2a-t-2-1-t nbsp-and y-3at-1-t

Dear student x=2at2t+1 #160;and #160;y=3at1+tdxdt=t+14at 2at2t+12=4at2+4at 2at2t+12and #160;dydt=1+t3a 3at1+t2=3a+3at 3att+12So, #160;dydx=2at2+4att+12 ...

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Standard XII

Mathematics

Differentiation to Solve Modified Sum of Binomial Coefficients

find second o...

Question

find second order derivative if: x=2a(t)^2/1+t and y=3at/1+t

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Solution

Dear student
$x = \frac{2 {at}^{2}}{t + 1} and y = \frac{3 at}{1 + t} \frac{dx}{dt} = \frac{(t + 1) 4 at - 2 {at}^{2}}{{(t + 1)}^{2}} = \frac{4 {at}^{2} + 4 at - 2 {at}^{2}}{{(t + 1)}^{2}} and \frac{dy}{dt} = \frac{(1 + t) 3 a - 3 at}{{(1 + t)}^{2}} = \frac{3 a + 3 at - 3 at}{{(t + 1)}^{2}} So, \frac{dy}{dx} = \frac{2 {at}^{2} + 4 at}{{(t + 1)}^{2}} \times \frac{{(t + 1)}^{2}}{3 a} = \frac{2 {at}^{2} + 4 at}{3 a} = \frac{2 at (t + 2)}{3 a} = \frac{2 t (t + 2)}{3} = \frac{2 t^{2} + 4 t}{3} \frac{d^{2} y}{{dx}^{2}} = \frac{4 t + 4}{3} = \frac{4 (t + 1)}{3}$
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find second order derivative if: x=2a(t)^2/1+t and y=3at/1+t

Dear student x=2at2t+1 and y=3at1+tdxdt=t+14at-2at2t+12=4at2+4at-2at2t+12and dydt=1+t3a-3at1+t2=3a+3at-3att+12So, dydx=2at2+4att+12×t+123a=2at2+4at3a=2att+23a=2tt+23=2t2+4t3d2ydx2=4t+43=4t+13 Regards