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. 

Implicit differentiation is useful when we have an equation containing a dependent variable , and and an independent variable , but we cannot solve to write  explicitly in terms of . 

Differentiate both sides simultaneously: 

The chain rule must be applied to the following derivatives: 

Noting the  is implied to be a function of , we have to first differentiate with respect to  and then differentiate  with respect to  Since we do not have an explicit definition for , we simply express its' derivative in the equation as , treating it as an "unknown," which needs to be solved for algebraically at a later step.  

For (1):

For (2):

Using these derivatives the main equation becomes: 

Now we simply solve for  and simplify as much as possible. Isolate all terms with a derivative onto one side of the equation and factor: 

On this problem we have to use chain rule, which is: 

So in this problem we let 

and 

.

Since 

and 

,

we can conclude that 

On this problem we have to use chain rule, which is: 

So in this problem we let 

and 

.

Since 

and 

,

we can conclude that