When do you chain rule

Answered by Michael O. Need help with Maths? One to one online tuition can be a great way to brush up on your Maths knowledge. Answered by Linfan S. If the last operation is to multiply or divide by a constant , the back up to the penultimate operation.

If the last operation on variable quantities is multiplication, use the product rule. If the last operation on variable quantities is division, use the quotient rule. If the last operation on variable quantities is applying a function, use the chain rule. Keep in mind that we can rewrite the expression and that doing so may change the appropriate differentiation rule. In general, this is how we think of the chain rule. We then differentiate the outside function leaving the inside function alone and multiply all of this by the derivative of the inside function.

In its general form this is,. The square root is the last operation that we perform in the evaluation and this is also the outside function. The outside function will always be the last operation you would perform if you were going to evaluate the function.

The derivative is then. In this case the outside function is the exponent of 50 and the inside function is all the stuff on the inside of the parenthesis. In this case we need to be a little careful. Recall that the outside function is the last operation that we would perform in an evaluation.

In this case if we were to evaluate this function the last operation would be the exponential. Therefore, the outside function is the exponential function and the inside function is its exponent. Remember, we leave the inside function alone when we differentiate the outside function. So, the derivative of the exponential function with the inside left alone is just the original function. Here the outside function is the natural logarithm and the inside function is stuff on the inside of the logarithm.

Again remember to leave the inside function alone when differentiating the outside function. There are two points to this problem. First, there are two terms and each will require a different application of the chain rule.

Deriving the Chain Rule When we have a function that is a composition of two or more functions, we could use all of the techniques we have already learned to differentiate it. The Chain and Power Rules Combined We can now apply the chain rule to composite functions, but note that we often need to use it with other rules.

Solution Because we are finding an equation of a line, we need a point. Hint Use the preceding example as a guide. Combining the Chain Rule with Other Rules Now that we can combine the chain rule and the power rule, we examine how to combine the chain rule with the other rules we have learned.

Solution First apply the product rule, then apply the chain rule to each term of the product. Hint Start out by applying the quotient rule. Composites of Three or More Functions We can now combine the chain rule with other rules for differentiating functions, but when we are differentiating the composition of three or more functions, we need to apply the chain rule more than once.

Then apply the chain rule several times. Proof of Chain Rule At this point, we present a very informal proof of the chain rule. Answer Finally, we put it all together. Key Concepts The chain rule allows us to differentiate compositions of two or more functions.