Question

Find (\partial w/\partial x)y,z if w=x^{2}+y-z+\sin t and x+y=t

Derivatives
ANSWERED
asked 2021-05-04

Find \(\displaystyle{\left(\partial\frac{{w}}{\partial}{x}\right)}{y},{z}\) if \(\displaystyle{w}={x}^{{{2}}}+{y}-{z}+{\sin{{t}}}\) and \(x+y=t\)

Answers (1)

2021-05-05

Step 1
To find the partial derivatives.
Step 2
given that
\(\displaystyle{w}={x}^{{{2}}}+{y}-{z}+{\sin{{t}}}\)
\(x+y=t\)
now
\(\displaystyle{w}={x}^{{{2}}}+{y}−{z}+{\sin{{\left({x}+{y}\right)}}}\)
\(\displaystyle{\frac{{\partial{w}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({x}^{{{2}}}+{y}-{x}+{\sin{{\left({x}+{y}\right)}}}\right)}\)
\(\displaystyle={2}{x}+{\cos{{\left({x}+{y}\right)}}}\)
\(\displaystyle{\frac{{\partial{w}}}{{\partial{y}}}}={\frac{{\partial}}{{\partial{y}}}}{\left({x}^{{{2}}}+{y}-{z}+{\sin{{\left({x}+{y}\right)}}}\right)}\)
\(\displaystyle={1}+{\cos{{\left({x}+{y}\right)}}}\)
\(\displaystyle{\frac{{\partial{w}}}{{\partial{z}}}}={\frac{{\partial}}{{\partial{z}}}}{\left({x}^{{{2}}}+{y}-{z}+{\sin{{\left({x},{y}\right)}}}\right)}\)
\(=-1\)

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