arildno said:Well, the total derivative of g is a matrix, whereas the total derivative of f is a vector. Together, they yield a vector.
Let's take a concrete example:
[tex]h(x,y)=f(u(x,y),v(x,y)), g(x,y)=(u(x,y),v(x,y))[/tex]
We have therefore, for example:
[tex]\frac{\partial{h}}{\partial{x}}=\frac{\partial{f}}{\partial{u}}\frac{\partial{u}}{\partial{x}}+\frac{\partial{f}}{\partial{v}}\frac{\partial{v}}{\partial{x}}[/tex]
This is then one of the two partial derivatives of h, the other being differentiation with respect to y.
It means the partial derivative of f with respect to u, of course. What else could it mean?frb said:what does
[tex]\frac{\partial{f}}{\partial{u}}[/tex]
mean?
It doesn't make sense. If f(u(a,b),v(a,b)) makes any sense then f is a function of u and v, not x and y. You must mean [tex]\frac{\partial f}{\partial u}[/tex] not [tex]\frac{\partial f}{\partial x}[/tex].Is the following formula correct?
[tex]\frac{\partial{h}}{\partial{x}}(a,b)=\frac{\partial{f}}{\partial{x}}(u(a,b),v(a,b))\frac{\partial{u}}{\partial{x}}(a,b)+\frac{\partial{f}}{\partial{y}}(u(a,b),v(a,b))\frac{\partial{v}}{\partial{x}}(a,b)[/tex]
The confusing chain differentiation rule, also known as the chain rule, is a mathematical rule used to find the derivative of a composite function. It is often used in calculus to find the rate of change of one variable with respect to another.
The chain rule can be confusing because it involves multiple functions and can be difficult to visualize. It also requires a good understanding of basic differentiation rules and algebraic manipulation.
To apply the chain rule, you must first identify the inner and outer functions of the composite function. Then, you can use the formula d/dx(f(g(x))) = f'(g(x)) * g'(x) to find the derivative.
Yes, the chain rule can be used for higher order derivatives. To find the second derivative, for example, you would apply the chain rule twice.
One tip for understanding the chain rule is to use the "inside-out" method, where you start with the inner function and work your way out. It is also helpful to practice with different examples and to break down the problem into smaller steps.