# Find the differential. a) z=\frac{1}{2}(e^{x^{2}+y^{2}}-e^{-x^{2}-y^{2}}) b) w=2z^{3}y\sin x

Find the differential.
a) $z=\frac{1}{2}\left({e}^{{x}^{2}+{y}^{2}}-{e}^{-{x}^{2}-{y}^{2}}\right)$
b) $w=2{z}^{3}y\mathrm{sin}x$
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limacarp4
Step 1
We have to find the differential
a) $z=\frac{1}{2}\left({e}^{{x}^{2}+{y}^{2}}-{e}^{-{x}^{2}-{y}^{2}}\right)$
Step 2
$⇒dz=\frac{1}{2}\left[{e}^{{x}^{2}+{y}^{2}}\left(2x\right)-{e}^{-{x}^{2}-{y}^{2}}\left(-2x\right)\right]dx+\frac{1}{2}\left[{e}^{{x}^{2}+{y}^{2}}\left(2y\right)-{e}^{-{x}^{2}-{y}^{2}}\left(-2y\right)\right]dy$
$⇒dz=x\left({e}^{{x}^{2}+{y}^{2}}+{e}^{-{x}^{2}-{y}^{2}}\right)dx+y\left({e}^{{x}^{2}+{y}^{2}}+{e}^{-{x}^{2}-{y}^{2}}\right)dy$

Jeremy Merritt
Step 3
b) $w=2{z}^{3}y\mathrm{sin}x$ $\left(\frac{\because d}{dx}\left({x}^{n}\right)=n{x}^{n-1}\right)$
then
$d\omega =6{z}^{2}y\mathrm{sin}xdz+2{z}^{3}\mathrm{sin}xdy+2{x}^{3}y\mathrm{cos}ndz$
$\left(\frac{\because d}{dx}\left(\mathrm{sin}x\right)=\mathrm{cos}x\right)$