$Find z(x, y), please help me with this calculus question$

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$Find z(x, y), please help me with this calculus question$

Given that $zx(x, y) = \sin(x) − ln(y)$ and that $z(y, y) = − cos(y)$, find $z(x, y).$

Calculus

Emmadavis

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We begin by integrating z_x(x,y) w.r.t. x.
$\int \sin(x) - \ln(y) dx=-\cos(x)-x\ln(y)+k(y)$ where k(y) is some function dependent on y. Note it acts like the constant k when integrating w.r.t. x since we treat y as a constant w.r.t. x.
Next we check $z(y,y)=-\cos(y)$. Using what we know about z we get $-\cos(y)=z(y,y)=-\cos(y)-y\ln(y)+k(y)$. Subtracting $\cos(y)$ from both sides and rearranging gives us $k(y)=y\ln(y)$. Taken all together we finally get:

$z(x,y)=-\cos(x)-x\ln(y)+y\ln(y)$.

Erdos

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$Find z(x, y), please help me with this calculus question$

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