If \(t = \sqrt{x} + 4\), then \(\left.\frac{dx}{dt}\right|_{t=4}\) is
- 4
- 0
- 8
- 16
The Correct Option is B
Approach Solution - 1
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 \(x\):
- Rewrite the equation: \(t = \sqrt{x} + 4\).
- Isolate \(\sqrt{x}\) by subtracting 4 from both sides: \(t - 4 = \sqrt{x}\).
- Square both sides to eliminate the square root: \(x = (t - 4)^2\).
Now, differentiate \(x\) with respect to \(t\):
- Apply the chain rule to differentiate: \(\frac{dx}{dt} = \frac{d}{dt}((t - 4)^2)\).
- Use the power rule: \(\frac{dx}{dt} = 2(t - 4) \cdot \frac{d(t-4)}{dt}\).
- 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 \(t = 4\):
- Substitute \(t = 4\) into the expression: \(\frac{dx}{dt} = 2(4 - 4)\).
- 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.
Approach Solution -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.
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Concepts Used:
Derivatives
Derivatives are defined as a function's changing rate of change with relation to an independent variable. When there is a changing quantity and the rate of change is not constant, the derivative is utilised. The derivative is used to calculate the sensitivity of one variable (the dependent variable) to another one (independent variable). Derivatives relate to the instant rate of change of one quantity with relation to another. It is beneficial to explore the nature of a quantity on a moment-to-moment basis.
Few formulae for calculating derivatives of some basic functions are as follows:
