Chain Rule Math Problems

Du dx 0 2 cos x sin x 2 sin x cos x.

Chain rule math problems. For problems 1 27 differentiate the given function. You must use the chain rule to find the derivative of any function that is comprised of one function inside of another function. The chain rule is a rule for differentiating compositions of functions. Differentiate the function y with respect to x.

G z 7 9z3 g z 9 z 3 7. R w csc 7w r w csc. The chain rule states formally that. Y u 6.

However we rarely use this formal approach when applying the chain rule to specific problems. For instance left x 2 1 right 7 is comprised of the inner function x 2 1 inside the outer function boxed phantom cdots 7. G x 2sin 3x tan x g x 2 sin. If we don t recognize that a function is composite and that the chain rule must be applied we will not be able to differentiate correctly.

The chain rule states dy dx dy du du dx in what follows it will be convenient to reverse the order of the terms on the right. Dy dx du dx dy du which in terms of f and g we can write as dy dx d dx g x d du f g x this gives us a simple technique which with some practice enables us to apply the chain rule directly key point. Differentiate the function u with respect to x. Y 3 1 8z y 1 8 z 3 solution.

R v 14v2 3v 2 r v 14 v 2 3 v 2. The general power rule the general power rule is a special case of the chain rule. Du dx sin 2x. Differentiate the square first leaving 3 x 1 unchanged.

Click here to return to the list of problems. F x 6x2 7x 4 f x 6 x 2 7 x 4 solution. The general power rule states that this derivative is n times the function raised to the n 1 th power times the derivative of the function. 7 w solution.

It is useful when finding the derivative of a function that is raised to the nth power. Dy dx 6u5 du dx 6 1 cos2x 5 sin 2x 6 sin 2x 1 cos2x 5. H t 9 2t t3 6 h t 9 2 t t 3 6. U 1 cos 2 x.

On the other hand applying the chain rule on a function that isn t composite will also result in a wrong derivative. This tutorial presents the chain rule and a specialized version called the generalized power rule. A few are somewhat challenging. Differentiation using the chain rule.

Usually the only way to differentiate a composite function is using the chain rule. Recall that with chain rule problems you need to identify the inside and outside functions and then apply the chain rule. The outer layer is the square and the inner layer is 3 x 1. In the following discussion and solutions the derivative of a function h x will be denoted by or h x.

G x 3 8x 11 g x 3 8 x 11. Differentiate f x 6x2 7x 4 f x 6 x 2 7 x 4. Most problems are average. Then differentiate 3 x 1.

For problems 1 51 differentiate the given function. Thus 2 3 x 1 3 6 3 x 1. H w 2 6 5w 8 h w 2 6 5 w 8.