Exercise
Given the differentiable function
z=\frac{y}{f(x^2-y^2)}
Prove the equation
\frac{1}{x}\cdot z'_x+\frac{1}{y}\cdot z'_y=\frac{z}{y^2}
Proof
Define
u=x^2-y^2
We get the function
z=\frac{y}{f(u)}=\frac{y}{f}
And the internal function
u(x,y)=x^2-y^2
We will use the chain rule to calculate the partial derivatives of z.
z'_x=z'_f\cdot f'_u\cdot u'_x=
=y\cdot \frac{-1}{f^2(u)}\cdot f'_u\cdot 2x=
=\frac{-2xyf'_u}{f^2(u)}
z'_y=\frac{1\cdot f(u)-y\cdot f'_u\cdot u'_y}{f^2(u)}=
=\frac{1}{f(u)}+\frac{2y^2f'_u}{f^2(u)}
We will put the partial derivatives in the left side of the equation we need to prove.
\frac{1}{x}\cdot z'_x+\frac{1}{y}\cdot z'_y=
=\frac{1}{x}\cdot \frac{-2xyf'_u}{f^2(u)}+\frac{1}{y}\cdot (\frac{1}{f(u)}+\frac{2y^2f'_u}{f^2(u)})=
=\frac{-2yf'_u}{f^2(u)}+\frac{1}{yf(u)}+\frac{2yf'_u}{f^2(u)}=
=\frac{1}{yf(u)}
Now, we will put the partial derivatives in the right side of the equation we need to prove.
\frac{z}{y^2}=\frac{\frac{y}{f(u)}}{y^2}=
=\frac{y}{f(u)}\cdot\frac{1}{y^2}=\frac{1}{yf(u)}
We got the same result in both sides as required.
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