Generalizing the chain rule

1. What is generalized chain rule?
2. How do you use generalized power rule?
3. How to do chain rule with multiple variables?
4. What is the chain rule in words?
5. What are the chain rule conditions?

What is generalized chain rule?

The General Version of the Chain Rule starts with a function f(x,y), where x and y are themselves functions x=x(s,t) and y=y(s,t) of two other variables s and t, so that the composition z = f(x(s,t),y(s,t)) is now a function of s and t.

How do you use generalized power rule?

The general power rule is a special case of the chain rule. It is useful when finding the derivative of a function that is raised to the nth power. The general power rule states that this derivative is n times the function raised to the (n-1)th power times the derivative of the function.

How to do chain rule with multiple variables?

Multivariable Chain Rules allow us to differentiate z with respect to any of the variables involved: Let x=x(t) and y=y(t) be differentiable at t and suppose that z=f(x,y) is differentiable at the point (x(t),y(t)). Then z=f(x(t),y(t)) is differentiable at t and dzdt=∂z∂xdxdt+∂z∂ydydt.

What is the chain rule in words?

The chain rule formula states that ^dy/_dx = ^dy/_du × ^du/_dx. In words, differentiate the outer function while keeping the inner function the same then multiply this by the derivative of the inner function.

What are the chain rule conditions?

We use the chain rule when differentiating a 'function of a function', like f(g(x)) in general. We use the product rule when differentiating two functions multiplied together, like f(x)g(x) in general. Take an example, f(x) = sin(3x).