The Chain Rule In Multivariable Calculus – Derivatives Of Composite Functions Explained

Chain Rule: f[g(t),h(t)]

df/dt=(δf/δx)(dx/dt)+(δf/δy)(dy/dt)

The chain rule is a powerful tool in calculus that allows us to find the derivative of a composite function. Specifically, the chain rule applies to functions of the form f[g(t),h(t)].

To apply the chain rule to this type of function, we must first identify the inner and outer functions. The inner functions are g(t) and h(t), and the outer function is f.

Next, we differentiate the outer function with respect to each of its variables. In other words, we find the partial derivative of f with respect to the first variable (holding the other variable constant) and multiply it by the derivative of the first inner function, g'(t). Then, we add to this the partial derivative of f with respect to the second variable (holding the other variable constant) multiplied by the derivative of the second inner function, h'(t).

In mathematical notation, the chain rule for a function of the form f[g(t),h(t)] is:

(d/dt) f[g(t),h(t)] = (∂f/∂x) g'(t) + (∂f/∂y) h'(t)

where x and y are the variables that correspond to the two inner functions.

It is important to note that this formula can be extended to functions of more than two variables in a straightforward manner. Also note that it is important to use partial derivatives when applying the chain rule to multivariable functions, as the variables we are not differentiating with respect to are treated as constants.

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