If x= 2+t 2 and y=t 3 find the value of (dy/dx)x(d 2 y/dx 2 )
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x=2+t²
differentiating both sides w.r.t. t we get,
dx/dt=2t
y=t³
differentiating both sides w.r.t. t we get,
dy/dt=3t²
∴, dy/dx=dy/dt÷dx/dt=3t²/2t=3t/2
∴, d²y/dx²
=d/dx(dy/dx)
=d/dx(3t/2)
=d/dt(3t/2)×dt/dx
=3/2×1/2t
=3/4t
∴, dy/dx×d²y/dx²=3t/2×3/4t=9/8
differentiating both sides w.r.t. t we get,
dx/dt=2t
y=t³
differentiating both sides w.r.t. t we get,
dy/dt=3t²
∴, dy/dx=dy/dt÷dx/dt=3t²/2t=3t/2
∴, d²y/dx²
=d/dx(dy/dx)
=d/dx(3t/2)
=d/dt(3t/2)×dt/dx
=3/2×1/2t
=3/4t
∴, dy/dx×d²y/dx²=3t/2×3/4t=9/8
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