Question

If \(t = \sqrt{x} + 4\), then \(\left.\frac{dx}{dt}\right|_{t=4}\) is

• A. 4
• B. 0
• C. 8
• D. 16

Answer 1

The Correct Option is B

To find \(\left.\frac{dx}{dt}\right|_{t=4}\), we need to first understand the given relationship \(t = \sqrt{x} + 4\). Our aim is to express \(x\) in terms of \(t\), and then differentiate with respect to \(t\).

First, solve for \(x\): 

1. Rewrite the equation: \(t = \sqrt{x} + 4\).
2. Isolate \(\sqrt{x}\) by subtracting 4 from both sides: \(t - 4 = \sqrt{x}\).
3. Square both sides to eliminate the square root: \(x = (t - 4)^2\).

Now, differentiate \(x\) with respect to \(t\):

1. Apply the chain rule to differentiate: \(\frac{dx}{dt} = \frac{d}{dt}((t - 4)^2)\).
2. Use the power rule: \(\frac{dx}{dt} = 2(t - 4) \cdot \frac{d(t-4)}{dt}\).
3. Since the derivative of \(t-4\) with respect to \(t\) is 1, we get: \(\frac{dx}{dt} = 2(t - 4)\cdot 1 = 2(t - 4)\).

Evaluate \(\frac{dx}{dt}\) at \(t = 4\):

1. Substitute \(t = 4\) into the expression: \(\frac{dx}{dt} = 2(4 - 4)\).
2. Evaluate: \(\frac{dx}{dt} = 2 \times 0 = 0\).

Therefore, the value of \(\left.\frac{dx}{dt}\right|_{t=4}\) is 0, which matches the correct answer.

Answer 2

\(x = (t – 4)^2\)
\(\frac{dx}{dt} = 2t - 8\)
For \(t=0\)
\(\frac{dx}{dt} = 2x^4 - 8\)
\(\frac{dx}{dt} = 0\)
So, the correct option is (B): 0.