To find the derivative, we can first rewrite the function to make it easier to take the chain rule.  Rewrite  as .  Now, like in any exponential function, the first factor of the derivative is the original exponential function.  So, the first factor of f'(x) will be .  Next, by the chain rule for derivatives, we must take the derivative of the exponent, which is why we rewrote the exponent in a way that is easier to take the derivative of.  So, the derivative of the exponent is , because the 1/2 and the 2 cancel when we bring the power down front, and the exponent of 1/2 minus 1 becomes negative 1/2.  The last factor of the derivative is  because in every derivative of an exponential function where the base is a number, we must multiply by the natural log of that base.  So, once you multiply all these factors together, the final answer is 

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get   The next step is to solve for y', so we put all terms containing y' on the left side of the equation and factor out a common y': .  To get y' alone, divide both sides by to get .

By the chain rule, we must first take the derivative of the outside function by bringing the power down front and reducing the power by one.  When we do this, we do not change the function that is in the parentheses, or the inside function.  That means that the first part of  will be .  Next, we must take the derivative of the inside function.  Its derivative is .  The chain rule says we must multiply the derivative of the outside function by the derivative of the inside function, so the final answer is .

Before we take the derivative of the logarithmic function, we can make it easier for ourselves by simplifying the equation to .  We can bring the exponent of 6 down in front of the natural log of x due to properties of logarithms.  Next, take the derivative of each term in terms of x.  Don't forget to multiply by y' each time you take the derivative of a term containing y!  When we do this, we should get  because the derivative of lnx is 1/x.  Next, solve for y' by multiplying both sides by y to get the final answer of .

To take the derivative of any exponential function, we repeat the exponential function in the derivative.  So, the first factor of the derivative will be .  Next, we have to take the derivative of the exponent using chain rule.  The derivative of the trigonometric function secx is secxtanx, so in terms of this problem its derivative is .  Since the angle has a scalar of 3, we must also multiply the entire derivative by 3.  So, the answer is .

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get .  The next step is to solve for y', so we put all terms containing y' on the left side of the equation: .  Next, factor out the y' from both terms on the left side of the equation so that we can solve for it: .  To get y' alone, divide both sides by  to get a final answer of  .

                          (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 

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get   The next step is to solve for y', so we put all terms containing y' on the left side of the equation: .  Next, factor out the y' from both terms on the left side of the equation so that we can solve for it:   To get y' alone, divide both sides by  to get .  To simplify even further, we can factor a 2 out of the numerator and denominator and cancel them.  So, the final answer is .

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 .