9 the chain rule homework solutions
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This imaginary computational process works every time to identify correctly what the inner and outer functions are. Click to return to the list of problems. Click to return to the list of problems. The key is to look for an inner function and an outer function. Click to see a detailed solution to problem 16. Different forms of chain rule: Consider the two functions f x and g x.

Step 1: Identify the inner and outer functions. Click to see a detailed solution to problem 13. The chain rule states formally that. Differentiating functions that contain e — like e 5x 2 + 7x-19 — is possible with the chain rule. We just left it in the derivative notation to make it clear that in order to do the derivative of the inside function we now have a product rule.

Thus, Now there are four layers in this problem. Click to return to the list of problems. Your comments and suggestions are welcome. Remember that a function raised to an exponent of -1 is equivalent to 1 over the function, and that an exponent of ½ is the same as a square root function. Here is the rest of the work for this problem.

There are two forms of the chain rule. The inner function is the one inside the parentheses: x 2 -3. Also note that again we need to be careful when multiplying by the derivative of the inside function when doing the chain rule on the second term. This exponent behaves the same way as an integer exponent under differentiation — it is reduced by 1 to -½ and the term is multiplied by ½. Step 4 Add the constant you dropped back into the equation. In the previous problem we had a product that required us to use the chain rule in applying the product rule.

Differentiating using the chain rule usually involves a little intuition. Differentiate the function by using the steps of chain rule. Label the function inside the square root as y, i. Just ignore it, for now. In this example, the inner function is 3x + 1. Click to return to the list of problems. Click to see a detailed solution to problem 6.

Knowing where to start is half the battle. The outer function in this example is 2 x. Click to return to the list of problems. In most cases, final answers are given in the most simplified form. Now, differentiating the final version of this function is a hopefully fairly simple Chain Rule problem. Click to see a detailed solution to problem 8.

Click to see a detailed solution to problem 4. Finish with the derivative of. Finish with the derivative of. Click to return to the list of problems. Example 2 Differentiate each of the following. In fact, this problem has three layers. Thus, Click to return to the list of problems.

In this case we need to be a little careful. Second, we need to be very careful in choosing the outside and inside function for each term. The derivative of x 4 — 37 is 4x 4-1 — 0, which is also 4x 3. Back in the on the definition of the derivative we actually used the definition to compute this derivative. Example 4 Differentiate each of the following. Step 2: Differentiate the inner layer 3 x. Instead, we invoke an intuitive approach.

There are four layers in this problem. Click to see a detailed solution to problem 20. First, there are two terms and each will require a different application of the chain rule. How can I tell what the inner and outer functions are? Sometimes these can get quite unpleasant and require many applications of the chain rule. Solution 1 quick, the way most people reason. Step 2 Differentiate the inner function, using. Differentiate them in that order.

This indicates that the function f x , the inner function, must be calculated before the value of g x , the outer function, can be found. The chain rule in calculus is one way to simplify differentiation. The second is more formal. In fact, this problem has three layers. There are four layers in this problem.