If y-t3-7t2-5 x-t2-4t-1find d2ydx2

y=t^3+7t^2+5 x=t^2+4t+1 ∴dydx=dydtdxdt=y'(t)x'(t)=3t^2+14t2t+4 ∴d^2ydx^2=ddx.dydx=ddtdydxdxdt =ddt(3t^2+14t2t+4)x'(t) =ddt(3t^2+14 ...

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Higher Order Derivatives

If y=t3+7t2...

Question

If $y = t^{3} + 7 t^{2} + 5$
$x = t^{2} + 4 t + 1$
find $\frac{d^{2} y}{d x^{2}}$

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Solution

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x = t^{2}, y = t^{3}

\frac{d^{2} y}{d x^{2}} =

x = t^{3} + t + 5

y = sin t

\frac{d^{2} y}{d x^{2}} =

t^{2} - 4 t + 1 = 0

t^{3} + \frac{1}{t^{3}}

x = \int_{0}^{y} \frac{d t}{\sqrt{1 + t^{2}}}

\frac{d^{2} y}{d x^{2}}

x = \sqrt{\frac{1 - t^{2}}{1 + t^{2}}}

y = \frac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{\sqrt{1 + t^{2}} + \sqrt{1 - t^{2}}},

\frac{d^{2} y}{d x^{2}}

t = 0

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