# How can evaluate partial derivatives, gradients, and directional derivatives at a point?

Question
Derivatives
How can evaluate partial derivatives, gradients, and directional derivatives at a point?

2020-11-30
Step 1
Partial derivative of a function is determined when a function having some variables is differentiated with respect to one of those variables assuming other remaining variables as constant and then substitute the given value of the variable in differentiated function
Step 2
Gradient of a function is determined as the value obtained when the coordinate of the point of interest is substituted in the partial differentiation of the function with respect to a variable.
Directional derivative of function at a point is determined as the ratio of the gradient vector of the point and the magnitude of the point.

### Relevant Questions

Find the second partial derivatives for the function $$\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{{4}}}-{3}{x}^{{{2}}}{y}^{{{2}}}+{y}^{{{2}}}$$ and evaluate it at the point(1,0).
Compute the gradient of the following functions and evaluate it at the given point P.
$$\displaystyle{g{{\left({x},{y}\right)}}}={x}^{{{2}}}-{4}{x}^{{{2}}}{y}-{8}{x}{y}^{{{2}}}.{P}{\left(-{1},{2}\right)}$$
consider the product of 3 functions $$\displaystyle{w}={f}\times{g}\times{h}$$. Find an expression for the derivative of the product in terms of the three given functions and their derivatives. (Remeber that the product of three numbers can be thought of as the product of two of them with the third
$$\displaystyle{w}'=$$?
Find the four second partial derivatives of the following functions.
$$\displaystyle{F}{\left({r},{s}\right)}={r}{e}^{{{s}}}$$
Find both first partial derivatives.
$$\displaystyle{f{{\left({x},{y}\right)}}}={2}{x}-{5}{y}+{3}$$
Find both first partial derivatives. $$\displaystyle{z}={\sin{{h}}}{\left({2}{x}+{3}{y}\right)}$$
Find both first partial derivatives. $$\displaystyle{z}={\sin{{\left({5}{x}\right)}}}{\cos{{\left({5}{y}\right)}}}$$
Find both first partial derivatives. $$\displaystyle{f{{\left({x},{y}\right)}}}={4}{x}^{{{3}}}{y}-{2}$$
$$\displaystyle{f{{\left({x},{y}\right)}}}={\ln{{\left({a}{x}+{b}{y}\right)}}}$$
Find both first partial derivatives. $$\displaystyle{z}={\sin{{\left({x}+{2}{y}\right)}}}$$