To solve this problem, we need the power rule, and the derivative of , which state:

After moving some things around with algebraic techniques, we obtain:

To solve this problem, we need the chain rule, the power rule, and the derivative of a constant.

Let's first rewrite the function in terms of a power:

Now we can use the chain rule, which states:

In this problem,  and 

To find  , we need the power rule which states:

To find , we need the power rule and the derivative of a constant which states:

Now, plugging these equations into the chain rule we obtain:

To solve this problem, we need the power rule and the chain rule, which state:

First let's apply the chain rule, which states:

In this problem,  and 

To find the , we need the power rule which states:

To find  we again need the power rule:

Now plugging these equations into the chain rule, we obtain:

To find the derivative of the function we must use the quotient rule.

It states that the derivative of 

 is .

The derivativae of  is  and the derivative of  is  as per the derivative rules.

Thus the final answer is 

The derivative is the function of the slope at any point of the given function. Thus we must find the derivative and then plug in 2 for x to get the slope of the tangent line.

The derivative of  is . The derivative of  is . So the derivative function is

plugging in  gives

.

The approach to this derivative is to realize that it is a function within a function and that we must use the chain rule.

The chain rule states the derivative of  is .

The derivative of  is  and the derivative of  is .

That makes the derivative of the function

.

To solve this problem, we need the power rule, the derivative of a constant, the product rule, and the chain rule.

Since our function is written as a product, we will first apply the product rule which states:

In this problem,  and .

To find  we need to use the power rule and the derivative of a constant which state that:

To find  we need to use the chain rule, the power rule, and the derivative of a constant. The chain rule states that:

In this problem,  and 

To find  we need the power rule.

To find  we need the power rule and the derivative of a constant once again.

Plugging these equations back into the chain rule, we obtain:

Now plugging this back into the product rule, we obtain:

After some simplification, we have:

To solve this problem, we need the power rule, the derivative of a constant, and the chain rule.

We will need to apply the chain rule twice in this problem, so let's do the first iteration.

The chain rule states:

In this problem,  and 

To find  , we need the power rule which states that:

To find  we need to again apply the chain rule, the power rule, and the derivative of a constant, which state:

To differentiate this equation, we can disregard the  which is added at the end, because it's derivative is equal to zero.

Here, we will use the following chain rule formula:

Where  and .

To find , we need the power rule.

To find , we need the power rule and the derivative of a constant.

Plugging these back into the chain rule formula, we obtain:

Now, plugging this equation back into the first iteration of the chain rule yields:

After some simplification and factoring we obtain:

Derivative of an exponential: 

Trigonometric derivative: 

Product rule: 

Note that u and v may represent large functions, and not just individual variables!

Now, approaching this problem, when taking the derivative with respect to one variable, treat the other variables as constant.

For the function 

Derivative with respect to x; note the use of the chain rule:

Derivative with respect to y:

Derivative with respect to z:

So the derivative of our function is the sum of these derivatives:

For this problem, we're asked to take the derivative of a function multiple times, each time with respect to a particular variable.

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant:

For the function

We'll make use of the following derivative properties:

Derivative of an exponential: 

Product rule: 

Note that u and v may represent large functions, and not just individual variables!

To begin finding   then:

Deriving with respect to x:

Now deriving the previous function with respect to y:

Note the use of the product rule!

Finally, deriving with respect to z: