(1)

An easier way to think about this:

Because  is a function of a function, we must apply the chain rule. This can be confusing at times especially for function like equation (1). The differentiation is easier to follow if you use a substitution for the inner function, 

Let,

                               (2)

So now equation (1) is simply, 

                               (3)

Note that  is a function of . We must apply the chain rule to find  , 

                            (4)

To find the derivatives on the right side of equation (4), differentiate equation (3) with respect to , then Differentiate equation (2) with respect to . 

Substitute into equation (4),  

                  (5)

Now use  to write equation (5) in terms of  alone: 

Here we use the chain rule: 

Let 

Then 

And 

Using the chain rule, we have

.

Hence, .

Notice that we could have also simplified  first by cancelling the natural log and the exponential function leaving us with just , thereby avoiding the chain rule altogether.

First, differentiate the outside of the parenthesis, keeping what is inside the same.

You should get  .

Next, differentiate the inside of the parenthesis. 

You should get .

Multiply these two to get the final derivative .

Use chain rule to solve this. First, take the derivative of what is outside of the parenthesis.

You should get .

Next, take the derivative of what is inside the parenthesis. 

You should get .

Multiplying these two together gives .

This is a chain rule derivative.  We must first start by taking the derivative of the outermost function.  Here, that is a function raised to the fifth power.  We need to take that derivative (using the the power rule).  Then, we multiply by the derivative of the innermost function:

This is a chain rule derivative.  We must first differentiate the natural log function, leaving the inner function as is. Recall:

Now, we must replace this with our function, and multiply that by the derivative of the inner function:

In this chain rule derivative, we need to remember what the derivative of the base function ().  The exponential function is a special function that always returns the original function.  In the chain rule application, we just need to multiply that by the derivative of the exponent.

.

Be careful with this derivative because it is a hidden chain rule! Let's start with rewriting the problem:

Now, it is easier to notice that it is a chain rule.  A chain rule involves differentiating the outermost function and then, multiplying by the derivative of the inner part.

This is a multiple chain derivative.  With chain derivatives, we always want to start on the outermost function (keeping the rest the same), and then, multiply by the inner function derivative until we have take the derivative of every part.