To take a partial derivative you treat all variables as constants except for the one youre taking the derivative of. So

\(\frac{\partial}{\partial x}f(x,y)\)

\(=\frac{δ}{δx}(x^{7}y)\)

\(y\frac{δ}{δx}(x^{7})\) [you factor out the y since its treated as constant in this case]

\(=7x^{6}y\)

\(b)\frac{δ}{δx}f(x,y)\)

\(=\frac{δ}{δx}(x^{7}y)\)

\(=x^{7}\frac{δ}{δx}(y)\) [you factor out the x^{7} since its treated as constant in this case]

\(x=7\)

\(\frac{\partial}{\partial x}f(x,y)\)

\(=\frac{δ}{δx}(x^{7}y)\)

\(y\frac{δ}{δx}(x^{7})\) [you factor out the y since its treated as constant in this case]

\(=7x^{6}y\)

\(b)\frac{δ}{δx}f(x,y)\)

\(=\frac{δ}{δx}(x^{7}y)\)

\(=x^{7}\frac{δ}{δx}(y)\) [you factor out the x^{7} since its treated as constant in this case]

\(x=7\)