Chain Rule

chutemedia | Jan 11, 2011 |

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At this point if you are studying calculus then you have probably heard of the chain rule. The chain rule is a general rule that can apply to any competent functions. There are of course more specific versions of this rule that can be applied to specific forms of functions. The most common and easy to understand is the power chain rule. Now usually we will start our study of the chain rule using the power chain rule except that we don't really define it as its own separate rule (because technically it isn't). In any case it is good to have a full understanding of the power chain rule. Essentially it says;

d/dx(u^n) = nu^(n-1) du/dx

See can see that this is just using the power rule on the outside function (u) and then multiplying it by the derivative of the inside function. Much like the normal chain rule. This law works great for when we have functions like;

f(x) = (sin(x) * e^x)^4

To find the derivative of the outside function is fairly easy. This is where user power chain rule. We just read everything that's in the brackets as a variable and use the power rule as you normally would. So we would get;

4*(sin(x) * e^x)^3

We can see that there's no tricks to doing this. It really is as easy as finding the derivative of any bracketed equations with the power rule.

Now to find the derivative of the inside function we're going to need a little more work. We can see here that we have the product of two functions so obviously we are going to have to use the product rule. Applying the product rule to the interior function we get;

cos(x) * e^x + e^x * sin(x) = e^x (sin(x) + cos(x))

Those you don't remember, to take the product rule we must take the derivative of the first times the second, plus the derivative of the second times the first. Now that we have both the inside and outside derivatives we can apply the chain rule as we normally would.

f'(x) = 4*(sin(x) * e^x)^3 * e^x (sin(x) + cos(x))

This is quite the messy equations so we will just leave it as is. You can attempt to make this simpler but you probably will not get very far. In any case that is a great example of where to use the power chain rule. You will see in the future that this is one of the most commonly used differentiation rules.