What is the derivative of e^x sin(3^(1/2x))?

May 2, 2018

$\frac{d}{\mathrm{dx}} \setminus {e}^{x} \sin \left({3}^{\frac{1}{2} x}\right) = \left(\frac{\ln 3 \setminus {3}^{\frac{x}{2}} \setminus \cos \left({3}^{\frac{x}{2}}\right)}{2} + \sin \left({3}^{\frac{x}{2}}\right)\right) {e}^{x} \setminus$

Explanation:

We seek:

$\frac{d}{\mathrm{dx}} \setminus {e}^{x} \sin \left({3}^{\frac{1}{2} x}\right)$

Using the product rule we have:

$\frac{d}{\mathrm{dx}} \setminus {e}^{x} \sin \left({3}^{\frac{1}{2} x}\right) = {e}^{x} \left(\frac{d}{\mathrm{dx}} \sin \left({3}^{\frac{1}{2} x}\right)\right) + \left(\frac{d}{\mathrm{dx}} {e}^{x}\right) \sin \left({3}^{\frac{1}{2} x}\right)$

By utilizing the property ${a}^{x} \equiv {e}^{x \ln a}$, and applying the chain rule, we have:

$\frac{d}{\mathrm{dx}} \sin \left({3}^{\frac{1}{2} x}\right) = \cos \left({3}^{\frac{1}{2} x}\right) \setminus \frac{d}{\mathrm{dx}} \left({3}^{\frac{1}{2} x}\right)$
$\text{ } = \cos \left({3}^{\frac{1}{2} x}\right) \setminus \frac{d}{\mathrm{dx}} \left({e}^{\frac{1}{2} x \ln 3}\right)$
$\text{ } = \cos \left({3}^{\frac{1}{2} x}\right) \setminus {e}^{\frac{1}{2} x \ln 3} \setminus \frac{d}{\mathrm{dx}} \left(\frac{1}{2} x \ln 3\right)$
$\text{ } = \cos \left({3}^{\frac{1}{2} x}\right) \setminus {3}^{\frac{1}{2} x} \setminus \left(\frac{1}{2} \ln 3\right)$

Thus:

$\frac{d}{\mathrm{dx}} \setminus {e}^{x} \sin \left({3}^{\frac{1}{2} x}\right) = {e}^{x} \setminus \cos \left({3}^{\frac{1}{2} x}\right) \setminus {3}^{\frac{1}{2} x} \setminus \left(\frac{1}{2} \ln 3\right) + {e}^{x} \setminus \sin \left({3}^{\frac{1}{2} x}\right)$

$\text{ } = \left(\frac{\ln 3 \setminus {3}^{\frac{x}{2}} \setminus \cos \left({3}^{\frac{x}{2}}\right)}{2} + \sin \left({3}^{\frac{x}{2}}\right)\right) {e}^{x} \setminus$