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:

The derivative is .

Now, substitute 1 for x.

This yields 7 as the slope of the tanget line at .

To solve this particular problem we will use the quotient rule to find the derivative. The quotient rule states to take the bottom function times the derivative of the top equation, subtract that product from the top function times the derivative of the bottom. Then divide this difference by the bottom function squared.

In mathematical terms,

Applying the quotient rule to our function gets,

.

This simplifies to , which further simplifies to .

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: 

Trigonometric derivatives: 

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 deritave with respect to y:

Finally, take the derivative with respect to x once more, and notice how terms without x go to zero:

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!

Looking for , begin by taking the derivative with respect to x:

Take the derivative once more, this time with respect to y:

Take the derivative with respect to y once more:

Finally, take the derivative with respect to x:

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:

Each of these functions gives a different case.  is a polynomial, and is consequently continuous and differentiable for the entire real line.   is not continuous at  since it is undefined.  yields a negative radicand at  and consequently a complex-valued answer (so it is not continuous since we are studying functions of real variables). Finally,  is continuous at  ( and the limit from the right equals zero, which is all that is necessary because the function is right handed from ). The derivative of , however is , which is undefined at , so  is not differentiable at .

This requires remembering the difference quotient and being careful with algebra. Much of the trouble that arises in calculating derivatives from the definition (and down the road, stickier integrals) is in faulty algebra. When plugging in  for  when calculating , be sure to FOIL. Also, when subtracting a sum of terms in parantheses, be sure to distribute the negative sign.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: 

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

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.

The function we're considering is

The derivatives follow as

The gradient is thus the sum of these partials, multiplied by their respective vectors:

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: 

Product rule: 

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

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

The gradient is then the sum of these multiplied by their respective vectors

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:

Derivative of an exponential: 

Trigonometric derivative: 

Product rule: 

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

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.

Looking at our function, take the derivative with respect to each variable

The complete derivative is then the sum of these: