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(d)/(dx)[sin((sqrt(e^(x)+a))/(2))]

$\frac{\mathrm{d}}{\mathrm{d} x}\left[\sin \left(\frac{\sqrt{\mathrm{e}^{x}+a}}{2}\right)\right]$

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Multiplication with rational exponents

Full solution

Q.

\frac{\mathrm{d}}{\mathrm{d} x}\left[\sin \left(\frac{\sqrt{\mathrm{e}^{x}+a}}{2}\right)\right]

Identify outer function: Identify the outer function and apply the chain rule. $\newline$ We start by recognizing that the outer function is $\sin(u)$ , where $u = (\sqrt{e^x + a})/2$ . The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$ .
Differentiate inner function: Differentiate the inner function $u = \frac{\sqrt{e^x + a}}{2}$ . Using the chain rule again, the derivative of $\sqrt{v}$ where $v = e^x + a$ is $\frac{1}{2}v^{-\frac{1}{2}}$ . Then, differentiate $v = e^x + a$ with respect to $x$ , which gives $e^x$ . So, $\frac{du}{dx} = \frac{1}{2}(e^x + a)^{-\frac{1}{2}} \cdot e^x$ .
Combine derivatives: Combine the derivatives using the chain rule. $\newline$ The derivative of the original function with respect to $x$ is $\frac{d}{dx}[\sin(u)] = \cos(u) \cdot \frac{du}{dx}$ . $\newline$ Substitute back for $u$ and $\frac{du}{dx}$ : $\newline$ $\frac{d}{dx}[\sin(\left(\sqrt{e^x + a}\right)/2)] = \cos(\left(\sqrt{e^x + a}\right)/2) \cdot \left(\frac{1}{2}\right)(e^x + a)^{-1/2} \cdot e^x$ .

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