Question

If x=5t and \(y=\frac5t,\) then \(\frac{d^2y}{dx^2}\) at t=1 is

• A. \(\frac25\)
• B. \(-\frac52\)
• C. \(\frac15\)
• D. \(-\frac25\)

Answer 1

The Correct Option is A

To find \(\frac{d^2y}{dx^2}\) when \(t=1\), we start by expressing \(y\) in terms of \(x\). Given: \(x=5t\) and \(y=\frac{5}{t}\).

First, express \(t\) in terms of \(x\): \(t=\frac{x}{5}\).

Now, substitute \(t=\frac{x}{5}\) into \(y\):
\(y=\frac{5}{\left(\frac{x}{5}\right)}=25 \cdot \frac{1}{x}=\frac{25}{x}.\)

Now differentiate \(y\) with respect to \(x\).

First derivative:

\(\frac{dy}{dx}=-\frac{25}{x^2}\)

Differentiate again to find the second derivative:

Second derivative:

\(\frac{d^2y}{dx^2}=\frac{50}{x^3}\)

Evaluate \(\frac{d^2y}{dx^2}\) at \(t=1\).

When \(t=1\), substitute \(x=5t\) to get \(x=5\).

Plug \(x=5\) into the second derivative:
\(\frac{d^2y}{dx^2}=\frac{50}{5^3}=\frac{50}{125}=\frac{2}{5}\).

Thus, the solution is \(\frac{2}{5}\).