erurnSopSoypegx

2021-12-05

Use the rules for derivatives to find the derivative of function defined as follows.
$y=10\cdot {2}^{\sqrt{x}}$

Gloria Lusk

Step 1: Consider the given
Function, $y=10\cdot {2}^{\sqrt{x}}$
Step 2: The objective
Is to find the derivative of the function.
Step 3: Concept used
Since, we know, $\frac{d}{dx}\left({a}^{g\left(x\right)}\right)=\left(\mathrm{ln}a\right){a}^{g\left(x\right)}{g}^{\prime }\left(x\right)$
Thus, here we will let $g\left(x\right)=\sqrt{x}⇒{g}^{\prime }\left(x\right)=\frac{1}{2\sqrt{x}}$
Step 4: Calculation
Hence, the required derivatives is as follows,
$\frac{dy}{dx}=10\cdot \frac{d}{dx}\left({2}^{\sqrt{x}}\right)$
$=10\left(\mathrm{ln}2\right){2}^{\sqrt{x}}\left(\frac{1}{2\sqrt{x}}\right)$
$=\frac{5}{\sqrt{x}}\left(\mathrm{ln}2\right){2}^{\sqrt{x}}$

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