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:

Note that for this problem, we're not told to take the derivative with respect to any particular variable, so it would be prudent to take the derivative with respect to both. Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: 

Trigonometric derivative: 

Product rule: 

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

For our function 

The derivative with respect to x is:

And the derivative with respect to y is:

The complete derivative is then the sum of these two:

Note that for this problem, we're not told to take the derivative with respect to any particular variable, so it would be prudent to take the derivative with respect to all three. Knowledge of the following derivative rules will be necessary:

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 our function

The derivative with respect to x is:

The derivative with respect to y is:

And the derivative with respect to z is:

The derivative of the function is then the sum of these:

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!

Since we're looking for  begin by taking the derivative with respect to x:

Now take the derivative with respect to y; be sure to utilize the product rule:

Notice that the derivative of the  term when treated as a constant is zero!

Finally, take the derivative with respect to z:

Note that for this problem, we're told to take the derivative with respect to only one variable, . This is what is known as taking a partial derivative.

Usually a partial derivative is denoted using the greek symbol delta as follows:

 where the partial is with respect to x. It may also be denoted as .

It is often useful when calculating the contribution of particular variables to a rate of change.

When taking a partial derivative, only treat the variable in question as an actual variable; all other variables treat as constants.

For this problem, we'll make use of the following derivative properties:

Derivative of an exponential: 

Trigonometric derivative: 

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

For the function

The derivative with respect to x is:

Note that since the natural log term had no  term, it was treated as a constant!

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.

Our function is  and we're asked to find 

Let's begin by finding , the derivative with respect to x; remember to treat the other variables as constant.

Notice how that last term goes to zero, since y and z are being treated as constant.

Next take the derivative with respect to y:

Again, another term goes to zero, since there is treated as a constant when taking the derivative with respect to y.

Deriving with respect to z then follows similarly:

Finally, taking the derivative once more with respect to x gives:

To find the derivative, simply use the power rule. To use the power rule, multiply the exponent by the coefficient and the result is the new coefficient. Meanwhile, the exponent gets knocked down by one. Do this for each part of the equation, and you will reach the answer.

Applying the power rule to each term in the function is as follows:

Recall that the derivative of a constant is zero.

Adding these terms back together results in our complete derivative:

Note that for this problem, we're told to take the derivative with respect to a particular variable. This is also known as taking a partial derivative, and it has uses in physics, engineering, and many other disciplines. A partial derivative is often denoted with the Greek character delta , or with a subscript denoting the variable being considered, i.e. .

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: 

Trigonometric derivative: 

Product rule: 

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

When taking a partial derivative for one particular variable, treat the other variables as constant.

The function we are given is

Make sure to utilize the product rule when taking the derivative!

Notice how the last term is zero, since it has no x terms and is treated as a constant!