This exercise gives mechanical practice calculating partial derivatives for polynomials.

If \(f(x,y) =4x+3y^2\), then \(\pp [f]{x} = \answer {4}\).

To compute the partial derivative with respect to \(x\), we will treat all other variables as constants and differentiate expressions that explicitly depend on \(x\) the same way we would before.
\[ \pp [f]{x} = \pp {x}\left [ 4x+3y^2 \right ] =\pp {x}\left [ 4x \right ] +\pp {x}\left [ 3y^2 \right ] = 4 + 0 \]
If \(f(x,y) = 12xy^5\), then \(\pp [f]{y} = \answer {60xy^4}\).
To compute the partial derivative with respect to \(y\), we will treat all other variables as constants and differentiate expressions that explicitly depend on \(x\) the same way we would before.
\[ \pp [f]{x} = \pp {x}\left [ 12xy^5 \right ] =12x \cdot \pp {x}\left [ y^5 \right ] = 12x \cdot 5y^4 \]
If \(f(x,y) = x^3+4x^2y^7+5y^6+7\), then \(\pp [f]{x} = \answer {3x^2+8xy^7}\) and \(\pp [f]{y} = \answer {28x^2y^6+30y^5}\).