Question

Write formulas for the indicated partial derivatives for the multivariable function.f(x,y)=7x^2+9xy+4y^3a)(delf)/(delx)b)(delf)/(dely)c)(delf)/(delx)|_(y=9)

Multivariable functions

Write formulas for the indicated partial derivatives for the multivariable function.
$$\displaystyle{f{{\left({x},{y}\right)}}}={7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}$$
a)$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}$$
b) $$\frac{\partial f}{\partial y}$$
c)$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}$$

2021-03-19
$$\displaystyle{f{{\left({x},{y}\right)}}}={7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}$$ (1)
a) Differentiating with respect to x partially, we have
$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}=\frac{\partial}{{\partial{x}}}{\left[{7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}\right]}$$
$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}={7}\cdot{2}{x}+{9}{y}+{0}$$
$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}={14}{x}+{9}{y}$$
b) Differentiating with respect to y partially, we have
$$\displaystyle\frac{{\partial{f}}}{{\partial{y}}}=\frac{\partial}{{\partial{y}}}{\left[{7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}\right]}$$
$$\displaystyle\frac{{\partial{f}}}{{\partial{y}}}={0}+{9}{x}+{4}\cdot{3}{y}^{{2}}$$
$$\displaystyle\frac{{\partial{f}}}{{\partial{y}}}={9}{x}+{12}{y}^{{2}}$$
c) $$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}={14}{x}+{9}\cdot{9}$$
$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}={14}{x}+{81}$$