In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be. The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining. The derivative of x is the constant function with value 1, and the derivative of f(g(x)) is determined by the chain rule. Therefore, we have: To express f′ as a function of an independent variable y, we substitute f(y) for x wherever it appears. Then we can solve for f′.‎Chain rule (probability) · ‎Integration by substitution · ‎Faà di Bruno's formula.


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Let chain rule differentiation function g t be the altitude of the car at time t, and let the function f h be the temperature h kilometers above sea level.

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For example, the altitude where the car starts is not known and the temperature on the mountain is not known. However, their derivatives are known: In this case we need to be chain rule differentiation little careful.

Recall that the outside function is the last operation chain rule differentiation we would perform in an evaluation. In this case if we were to evaluate this function the last operation would be the exponential.

Simple examples of using the chain rule - Math Insight

Therefore, the outside function is the exponential function and the inside function is its exponent. So, the derivative of the exponential function with the inside left alone is just the original function.

First, there are two terms and each will require a different application of the chain rule. Second, we need to be very careful in choosing the chain rule differentiation and inside function for each term.

Chain rule (video) | Khan Academy

Example 3 Differentiate chain rule differentiation of the following. We will be assuming that you can see our choices based on the previous examples and the work that we have shown.

Example 4 Differentiate each of the following. Let us find the derivative of We have.

Chain rule (article) | Khan Academy

Then the Chain rule implies that f' x exists and In fact, this is a particular case of the following formula The following formulas come in handy in many areas of techniques of integration.

More formulas for derivatives can be found in our section of chain rule differentiation.

Find the derivative of. This right over here is the derivative.

Chain rule

Chain rule differentiation taking the derivative of, we're taking the derivative of sine of x squared. That's what we were taking the derivative of with respect to sine of x, with respect to sine of x. And then we're multiplying that times the derivative of sine of x, chain rule differentiation derivative of sine of x with respect to, with respect to x.


And this is where it might start making a chain rule differentiation bit of intuition. You can't really treat these differentials, this d whatever, this dx, this d sine of x, chain rule differentiation a number.

And you really can't, this notation makes it look like a fraction because intuitively that's what we're doing. But if you were to treat 'em like fractions, then you could think about canceling that and that.

And once again, this isn't a rigorous thing to do, but it can help with the intuition.