First slide

Derivatives

Question

$If y = e^{\sqrt{x}} + e^{- \sqrt{x}}, then \frac{dy}{dx} is equal to$

Moderate

A

$\frac{e^{\sqrt{x}} - e^{- \sqrt{x}}}{2 \sqrt{x}}$
B

$\frac{e^{\sqrt{x}} - e^{- \sqrt{x}}}{2 x}$
C

$\frac{1}{2 \sqrt{x}} \sqrt{y^{2} - 4}$
D

$\frac{1}{2 \sqrt{x}} \sqrt{y^{2} + 4}$

Solution

$\begin{array}{l} \frac{dy}{dx} = \frac{e^{\sqrt{x}}}{2 \sqrt{x}} - \frac{e^{- \sqrt{x}}}{2 \sqrt{x}} = \frac{e^{\sqrt{x}} - e^{- \sqrt{x}}}{2 \sqrt{x}} \\ = \frac{\sqrt{{(e^{\sqrt{x}} + e^{- \sqrt{x}})}^{2} - 4}}{2 \sqrt{x}} \\ = \frac{\sqrt{y^{2} - 4}}{2 \sqrt{x}} \end{array}$

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