Can anyone explain how the "Chain rule" works in calculus? Please no links...?

Question

I have seen a lot of the links and still have trouble understanding it...so if you can teach it in simple ways of understanding it...it would help a lot of us trying to learn it!

The link you provided me with will be a great help...thanks!

Answer 1

http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/chainruledirectory/ChainRule.html

I know you said no links, but that helps more than my explanation would. Please try the practice problems etc etc and if you have more specific roblems with it, ask those instead of the general conceptual question.

Answer 2

Calculus is a factor of a better branch of arithmetic referred to as "diagnosis", that's, by utilising definition, the learn of infinity and bounds. What you're doing once you're taking "shortcuts" is taking the habit of the function and utilising it to entice conclusions approximately concepts at a single ingredient (like the by-product), or over the finished era (the crucial). as an occasion, enable's assume you opt to be certain how some distance a horse on the track has run. ideally, you're able to easily multiply it quite is speed by utilising the time it ran. regrettably, this would not artwork nicely in prepare, because of the fact it runs at many distinctive speeds from the time it leaves the gate to the time it crosses the end line. What you're able to could desire to do then, is multiply each distinctive speed by utilising the quantity of time it become working at that speed, then upload 'em all up. The smaller the periods of time you think approximately, the greater precise your answer would be. So in case you think of approximately on the spot speed (that's speed at a single 2d in time), your answer would be one hundred% finished. A mathematical merchandise like that's referred to as an crucial, of direction. This, lower back, is the undemanding concept of a shrink, examining the habit of the function (distance) over smaller and smaller periods (time) to derive concepts a pair of single ingredient (on the spot speed).

Answer 3

OK, I'll try. The key is that you need to identify an "outside" function and an "inside" function. For example in sin(x^3), "sin" is the outside function and "cubed" is the inside function. On the other hand, in (cosx)^2, "squared" is the outside function and "cos" is the inside function.

Basically the chain rule says you take the derivative of the outside TIMES the derivative of the inside. The only trick is that when you do the derivative of the outside, you copy the inside stuff with it.

As an example, the derivative of sin(x^3) is cos(x^3)*3x^2 because the derivative of sin is cos and the derivative of x^3 is 3x^2. Most books will write the exponent part first and show the answer as 3x^2cos(x^3).

Another example: the derivative of (cosx)^2 is 2cosx*sinx because the derivative of something squared is 2 times that something and the derivative of cos is sin.

If you have trouble identifying the outside and inside functions, it may help that "inside" usually means one of these things:
--the thing you're taking a trig function of
--what is under a root
--the exponent on "e"
--what is inside parentheses
--what is on the bottom of a fraction

Again the main thing to remember is derivative of outside times derivative of inside