Question #32432

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Answer 1 · 2016-12-11T19:42:56

$\frac{d y}{d u} = \frac{1}{2} e^{\frac{x}{2}} = \frac{1}{2} \sqrt{e^{x}}$ $(dy)/(du)=1/2e^(x/2)=1/2sqrt(e^x)$

rewrite the function as a power of $e$ $""e$

$y = \sqrt{e^{x}} = {(e^{x})}^{\frac{1}{2}}$ $y=sqrt(e^x)=(e^x)^(1/2)$

$y = e^{\frac{x}{2}}$ $y=e^(x/2)$

$\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$ $(dy)/(dx)=(dy)/(du)xx(du)/(dx)$

$u = \frac{x}{2} \Rightarrow \frac{d u}{d x} = \frac{1}{2}$ $u=x/2=>(du)/(dx)=1/2$

$y = e^{u} \Rightarrow \frac{d y}{d u} = e^{u}$ $y=e^u=>(dy)/(du)=e^u$

$\frac{d y}{d x} = e^{u} \times \frac{1}{2}$ $(dy)/(dx)=e^uxx1/2$

tidying up, and substituting back for $u$ $" "u$

$\frac{d y}{d u} = \frac{1}{2} e^{\frac{x}{2}} = \frac{1}{2} \sqrt{e^{x}}$ $(dy)/(du)=1/2e^(x/2)=1/2sqrt(e^x)$