Differential coefficient of x is - Vidyadip

MCQ

Differential coefficient of $\sqrt{\sec \sqrt{x}}$ is

A
$\frac{1}{4 \sqrt{x}} \sec \sqrt{x} \sin \sqrt{x}$
✓
$\frac{1}{4 \sqrt{x}}(\sec \sqrt{x})^{3 / 2} \cdot \sin \sqrt{x}$
C
$\frac{1}{2} \sqrt{x} \sec \sqrt{x} \sin \sqrt{x}$
D
$\frac{1}{2} \sqrt{x}(\sec \sqrt{x})^{3 / 2} \cdot \sin \sqrt{x}$

✓

Answer

Correct option: B.

$\frac{1}{4 \sqrt{x}}(\sec \sqrt{x})^{3 / 2} \cdot \sin \sqrt{x}$

Let $y=\sqrt{\sec \sqrt{x}}$
Differentiating $\text{w.r.t. x}$, we get
$\frac{d y}{d x}=\frac{1}{2 \sqrt{\sec \sqrt{x}}} \cdot \sec \sqrt{x} \cdot \tan \sqrt{x} \cdot \frac{1}{2 \sqrt{x}}$
$=\frac{1}{4 \sqrt{x}}(\sec \sqrt{x})^{1 / 2} \frac{\sin \sqrt{x}}{\cos \sqrt{x}}$
$=\frac{1}{4 \sqrt{x}}(\sec \sqrt{x})^{3 / 2} \cdot \sin \sqrt{x}$

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