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$\frac{\ⅆ}{\ⅆ x}{\ⅇ}^{\sqrt{x}}$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Differentiation Steps}\\ & & \frac{\ⅆ}{\ⅆx}{\ⅇ}^{\sqrt{x}}\\ \text{▫}& & \text{1. Apply the}\mathbf{\text{chain}}\text{rule to the term}{\ⅇ}^{\sqrt{x}}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{chain}}\text{rule}\\ & & \frac{\ⅆ}{\ⅆx}f\left(g\left(x\right)\right)=\mathrm{f\text{'}}\left(g\left(x\right)\right)\frac{\ⅆ}{\ⅆx}g\left(x\right)\\ & \text{◦}& \text{Outside function}\\ & & f\left(v\right)={\ⅇ}^v\\ & \text{◦}& \text{Inside function}\\ & & g\left(x\right)=\sqrt{x}\\ & \text{◦}& \text{Derivative of outside function}\\ & & \frac{\ⅆ}{\ⅆv}f\left(v\right)={\ⅇ}^v\\ & \text{◦}& \text{Apply composition}\\ & & \mathrm{f\text{'}}\left(g\left(x\right)\right)={\ⅇ}^{\sqrt{x}}\\ & \text{◦}& \text{Derivative of inside function}\\ & & \frac{\ⅆ}{\ⅆx}g\left(x\right)=\frac{1}{2\sqrt{x}}\\ & \text{◦}& \text{Put it all together}\\ & & \frac{\ⅆ}{\ⅆx}f\left(g\left(x\right)\right)\frac{\ⅆ}{\ⅆx}g\left(x\right)={\ⅇ}^{\sqrt{x}}\cdot \left(\frac{1}{2\sqrt{x}}\right)\\ & & \text{This gives:}\\ & & \frac{{\ⅇ}^{\sqrt{x}}}{2\sqrt{x}}\end{array}$

Table 2.6.1(a)   Annotated stepwise solution

Define the function $f$ 

$f\left(x\right)\={\ⅇ}^{\sqrt{x}}$$\stackrel{\text{assign as function}}{\to }$$f$

Solution #1

$f\prime \left(x\right)$ = $\frac{1}{2}\frac{{\ⅇ}^{\sqrt{x}}}{\sqrt{x}}$$$

Solution #2

$\frac{\ⅆ}{\ⅆ x} f\left(x\right)$ = $\frac{1}{2}\frac{{\ⅇ}^{\sqrt{x}}}{\sqrt{x}}$$$

Solution #3

$f\left(x\right)$$\stackrel{\text{differentiate w.r.t. x}}{\to }$$\frac{1}{2}\frac{{\ⅇ}^{\sqrt{x}}}{\sqrt{x}}$