"da/dx" and "dc/da" are NOT fractions so it is not correct to say that the "da" cancels!pivoxa15 said:Homework Statement
(d/dx)c=(d/da)c(da/dx)
where c=f(a(x))
The Attempt at a Solution
It seems correct because the da cancels but how do you get this from first principles?
radou said:You may want to investigate the chain rule.
Are you saying you have never heard of the chain rule? You are being asked to prove the chain rule!pivoxa15 said:Could you possibly be a little bit more specific?
HallsofIvy said:"da/dx" and "dc/da" are NOT fractions so it is not corret to say that the "da" cancels!
Yes, because the derivative is the limit of a fraction, you can always treat them "like" a fraction- that's one of the advantages of the dy/dt notation over f '. And, in fact, it is motivation for defining the "differentials", dx and dy= f '(x) dx.pivoxa15 said:But physicists and applied mathematicians like to treat them as fractions in the limit. Is it okay to treat them as fractions and specify "in the limit"?
The chain rule is a formula used in calculus to find the derivative of a composition of functions. It states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
To apply the chain rule, you first identify the outer function and the inner function. Then, you take the derivative of the outer function and evaluate it at the inner function. Finally, you multiply this result by the derivative of the inner function.
This notation represents the chain rule in differentiation. The "d/dx" indicates that we are taking the derivative with respect to x. The "c" represents the composite function, "f" represents the outer function, "a(x)" represents the inner function, and "da/dx" represents the derivative of the inner function.
The chain rule is important because it allows us to find the derivative of more complex functions by breaking them down into simpler functions. This is especially useful when dealing with functions that are composed of multiple layers of functions, making it difficult to find the derivative using other methods.
Yes, for example, if we have the function f(x) = sin(x^2), we can use the chain rule to find its derivative. First, we identify the outer function as sin(x) and the inner function as x^2. The derivative of the outer function is cos(x), and the derivative of the inner function is 2x. Therefore, the derivative of f(x) is f'(x) = cos(x^2) * 2x.