What is the chain rule of second derivative?

Take the second derivative (use the chain rule, product rule, and chain rule, respectively, in that order): d2yd2x=ddx(df(u)dududx)=df(u)dud2udx2+d2f(u)du2(dudx)2.

What is the multivariable chain rule?

generalized chain rule the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables intermediate variable given a composition of functions (e.g., f(x(t),y(t))), the intermediate variables are the variables that are independent …

When to use the chain rule for multivariable functions?

I think you’re mixing up the chain rule for single- and multivariable functions. For the single variable case, For a two-variable function things are more complicated. Suppose we have a function f (x,y) where x and y are themselves functions x (r,t) and y (r,t). As you stated,

How to calculate multivariate chain rule and second order partials?

Equally, d dt(∂f ∂ydy dt) = ∂f ∂yd2y dt2 + (dy dt)2∂2f ∂y2 + dy dt dx dt ∂2f ∂x∂y so adding gives The important thing to remember is that ∂f / ∂x and friends are all still just functions, in the same way that f itself is, albeit with rather more complicated symbols.

When to use the chain rule for second derivatives?

Suppose we have a function f (x,y) where x and y are themselves functions x (r,t) and y (r,t). As you stated, To make things simpler, let’s just look at that first term for the moment. The tricky part is that ∂ f ∂ x is still a function of x and y, so we need to use the chain rule again.

Which is the chain rule for a composite function?

Recall that the chain rule for the derivative of a composite of two functions can be written in the form d dx(f(g(x))) = f′(g(x))g′(x). In this equation, both f(x) and g(x) are functions of one variable. Now suppose that f is a function of two variables and g is a function of one variable.