The Chain Rule By Dr. Julia Arnold

Rule 7: The Chain Rule If h(x) = g(f(x)), then h’(x) = g’(f(x))f’(x). The Chain Rule deals with the idea of composite functions and it is helpful to think about an outside and an inside function when using The Chain Rule. In other words: The derivative when using the Chain Rule is the derivative of the outside leaving the inside unchanged times the derivative of the inside.

Let’s consider the function h(x) = f(g(x)) where f(x) = x4 and g(x) = x + 2, then h(x)= f(g(x))= (x+2)4. We could find the derivative by expanding (x + 2)4 and then using the Power Rule. h(x)=f(g(x)) = x4 +8x3 + 24x2 + 32x +16 So then, h’(x) =[f(g(x))]’ = 4x3 + 24x2 +48x +32 This is a perfectly acceptable way to find the derivative, but multiplying out the binomial can be time consuming. So, now that we know the derivative, let’s find the derivative using the Chain Rule.

Using the Chain Rule, it will be helpful to identify the outside and inside before beginning. h(x) = (x+2)4. The outside = ( )4 and the inside = x+2. Can you identify these? Now using the chain rule: Derivative = derivative of outside leaving inside * the derivative of the inside. h’(x) = (4(x+2)3)*(1) h’(x)= 4(x+2)3

Now you may be saying to yourself “Wait those two derivatives were not the same.” When we found the derivative without the Chain Rule we came up with h’(x) =[f(g(x))]’ = 4x3 + 24x2 +48x +32. When we found the derivative using the Chain Rule we came up with h’(x)= 4(x+2)3. To see that these two are the same you simply need to expand the second one. Now, we would not normally expand that derivative. It was done only for the purpose of verifying that the derivatives were the same.

Another representation of the Chain Rule is: If u = g(x) and f(u) = y then where and Comparing the two representations:

Another example: Find the derivative of It is helpful to identify the outside function and the inside function. In this example, the outside function is the cube, and the inside function is x2 +3. The chain rule says take the derivative of the outside function leaving the inside function unchanged and then multiply by the derivative of the inside function. The derivative of the inside using the Power Rule The derivative of the outside leaving the inside unchanged

Finally we need to simplify the answer. So the solution to finding the derivative of is .

Example 3: Find the derivative of The outside function is the cube root function and the inside function is . First rewrite the function with rational exponent: To find the derivative of the outside, do the Power Rule: Note: This is just HOW we are finding the derivative of the outside function.

Now let’s look at the actual derivative using the Chain Rule. The derivative of the outside leaving the inside unchanged The derivative of the inside Now do a little simplification: Multiply the 1/3 and the 6x.

Example 4: Find the derivative of: This could be done by the quotient rule, but the numerator is a constant whose derivative is 0. Since the numerator does not contain a variable we could just do a rewrite and use the chain rule. Since 2 is a constant the derivative of this expression will be determined by the chain rule. derivative of the inside Don’t mess with the inside!!!

The result needs some simplifying: Multiply the constants together.

Example 5: Find the derivative of: The outside is , the inside is . First rewrite the square root as the exponent 1/2. To find the derivative we will need to use the Chain Rule and when we multiply by the derivative of the inside we will need to use the Quotient Rule to find the derivative of .

Now we need to do the Quotient Rule on the inside. Continued…

Simplifying more … Move last term to front and combine with 1/2, also use rule of negative exponent to make positive exponent. Use radical sign for 1/2 exponent.

In this problem you are being asked to multiply or expand the binomial In this problem you are being asked to multiply or expand the binomial. It is not asking for a derivative. You can use FOIL to find the product First Outside Inside Last

Suppose you did want to find the derivative of this problem. Your first step would be to rewrite the radical as an exponential Step 1 Step 2 would be to decide what rule applies to this problem. Power Rule Chain Rule Product Rule Click on the arrow of the answer you would pick.

If you picked the chain rule, you are correct.

Continuing Algebra Stuff Now to simplify using our algebra skills.

Rationalizing the denominator and Reducing. Yeah! Its finished.

You may have noticed in the last example that when you are using several rules for differentiating in combination, the algebra can get very complicated. That is why it is essential that before you begin a problem you have determined HOW you are going to find the derivative and that you KNOW the rules you are using.

Sorry, if you picked the power rule you are close because when you perform the chain rule on the outside, you will be using the power rule. Go back and pick again.

Sorry, if you picked the product rule, we don’t really have a product unless you write it as a product Product To use the product rule you would have to start with this. Go back and choose again.