Given the differentiable function
Prove the equation
When we have a function and in its parentheses there is a complex expression instead of a simple variable, we will define a new variable like this
We get the function
Now we will use the chain rule to calculate the partial derivatives of z.
We will calculate the partial derivatives of t.
We put the results in the partial derivatives of u and get
u'_x=f'_t\cdot t'_x=f'_t\cdot 2xz
u'_y=f'_t\cdot t'_y=f'_t\cdot (-z)
u'_z=f'_t\cdot t'_z=f'_t\cdot (x^2-y)
We will put the partial derivatives in the left side of the equation we need to prove.
=xf'_t\cdot 2xz+2yf'_t\cdot (-z)-2zf'_t\cdot (x^2-y)=
Hence, we get
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