d²y/dx² is just the derivative of dy/dx, when dy/dx is defined as a function of x. But in this case, it is defined as a function of t, so we need to use chain rule x = e^t dx/dt = e^t ------> dt/dx = e^(−t) y = te^(−t) dy/dt = e^(−t) − t e^(−t) = −e^(−t) (t − 1) dy/dx = (dy/dt) / (dx/dt ) = −e^(−t) (t − 1) / e^t = −e^(−2t) (t − 1) So far, so good -------------------- d²y/dx² = d/dx (dy/dx) . . . . . . = d/dx (−e^(−2t) (t − 1)) . . . . . . = d/dt (−e^(−2t) (t − 1)) * dt/dx -----> chain rule . . . . . . = e^(−2t) (2t − 3) * e^(−t) . . . . . . = e^(−3t) (2t − 3)
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