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Chapter 6: Problem 113

If \(x=a t^{2}, y=2 a t\), find \(\frac{d^{2} y}{d x^{2}}\)

Short Answer

Expert verified
Hence, the second derivative \(\frac{d^2 y}{d x^2} = -1/ 2ax\)

Step by step solution

01

Finding the first derivative dy/dx

Given \(y=2at\), the first derivative \(dy/dt = 2a\). Likewise, from \(x = at^2\), find the first derivative \(dx/dt = 2at\). Dividing \(dy/dt\) by \(dx/dt\) will give \(dy/dx = 2a / 2at = 1/t\)
02

Finding the second derivative \(d^2y/dx^2\)

Now, taking the derivative of dy/dx using the chain rule: \(d (dy/dt) / dx = d(1/t) / dt * dt/dx = -1/t^2 * dt/dx\). Here, replace \(dt/dx\) with 1/ \(dx/dt\) obtained in step 1. We get \(-1/t^2 * 1/2at = -1/ 2at^3\)
03

Substituting \(x = at^2\)

Now, replace \(t^2\) with \(x/a\), the second derivative \(d^2y/dx^2 = -1/ 2ax\)

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