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$\frac{\mathrm{d}}{\mathrm{d} x}\left[\sin \left(\frac{\sqrt{\mathrm{e}^{x}+a}}{...

Dec 22, 2023

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

Solution by Steps

step 1

Apply the chain rule for differentiation, which states that the derivative of a composite function

f(g(x))

f'(g(x)) \cdot g'(x)

step 2

Let

u = \frac{\sqrt{e^x + a}}{2}

, then differentiate

\sin(u)

with respect to

u

to get

\cos(u)

step 3

Differentiate

u = \frac{\sqrt{e^x + a}}{2}

with respect to

x

using the chain rule:

u' = \frac{1}{2} \cdot \frac{1}{2\sqrt{e^x + a}} \cdot e^x

step 4

Combine the results from steps 2 and 3 to get the derivative of the original function:

\frac{d}{dx}[\sin(u)] = \cos(u) \cdot u'

step 5

Substitute

u

back into the derivative to get the final answer:

\frac{d}{dx}[\sin(\frac{\sqrt{e^x + a}}{2})] = \cos(\frac{\sqrt{e^x + a}}{2}) \cdot \frac{1}{2} \cdot \frac{1}{2\sqrt{e^x + a}} \cdot e^x

Answer

\frac{d}{dx}\left[\sin \left(\frac{\sqrt{e^x + a}}{2}\right)\right] = \cos\left(\frac{\sqrt{e^x + a}}{2}\right) \cdot \frac{e^x}{4\sqrt{e^x + a}}

Key Concept

Chain rule for differentiation

Explanation

The derivative of the sine of a function is the cosine of that function times the derivative of the inner function, applying the chain rule.