In[1]:= f[x_, y_] := {-2 x + 2 x^2, -3 x + y + 3 x^2};
df = D[f[x, y], {{x, y}}]
Out[2]= {{-2 + 4 x, 0}, {-3 + 6 x, 1}}
In[3]:= df /. {x -> 1, y -> 2}
Out[3]= {{2, 0}, {3, 1}}In[4]:= D[f[x, y], {{x, y}}] /. {x -> 1, y -> 2}
Out[4]= {{2, 0}, {3, 1}}A derivative is a mathematical concept that represents the rate of change of a function at a particular point. It is important because it allows us to analyze and understand the behavior of a function, such as its slope and curvature, which have real-world applications in fields like physics, economics, and engineering.
To evaluate a derivative at a point in Mathematica, you can use the built-in function "D" and specify the function and the point at which you want to evaluate the derivative. For example, to evaluate the derivative of f(x) at x=3, you would use the command "D[f[x],x]/.x->3".
Yes, Mathematica has the ability to compute derivatives of complex functions using the same syntax as for real functions. However, it is important to note that the result may also be complex, so it is necessary to use appropriate functions to visualize or manipulate the result.
To plot the derivative of a function in Mathematica, you can use the "Plot" function and specify the derivative as the function to be plotted. For example, to plot the derivative of f(x), you would use the command "Plot[D[f[x],x], {x, a, b}]", where "a" and "b" are the range of values for the x-axis.
Yes, Mathematica has built-in functions such as "Solve" and "NSolve" that can find the critical points and inflection points of a function. You can also use the "FindRoot" function to numerically find these points. Additionally, you can use the "Plot" function to visualize these points on a graph.