"Find the differential. a) z=12(ex2+y2−e−x2−y2) b) w=2z3ysin⁡x" was answered by our experts

fertilizeki

Answered question

2021-12-13

Find the differential.
a)

z = \frac{1}{2} (e^{x^{2} + y^{2}} - e^{- x^{2} - y^{2}})

w = 2 z^{3} y \sin x

limacarp4

Beginner2021-12-14Added 39 answers

Step 1
We have to find the differential
a)

z = \frac{1}{2} (e^{x^{2} + y^{2}} - e^{- x^{2} - y^{2}})

Step 2

\Rightarrow d z = \frac{1}{2} [e^{x^{2} + y^{2}} (2 x) - e^{- x^{2} - y^{2}} (- 2 x)] d x + \frac{1}{2} [e^{x^{2} + y^{2}} (2 y) - e^{- x^{2} - y^{2}} (- 2 y)] d y

\Rightarrow d z = x (e^{x^{2} + y^{2}} + e^{- x^{2} - y^{2}}) d x + y (e^{x^{2} + y^{2}} + e^{- x^{2} - y^{2}}) d y

Jeremy Merritt

Beginner2021-12-15Added 31 answers

Step 3
b)

w = 2 z^{3} y \sin x

(\frac{∵ d}{d x} (x^{n}) = n x^{n - 1})

then

d ω = 6 z^{2} y \sin x d z + 2 z^{3} \sin x d y + 2 x^{3} y \cos n d z

(\frac{∵ d}{d x} (\sin x) = \cos x)

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