To consider finding the slope, let's discuss the topic of the gradient.

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: 

Derivative of a natural log: 

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.

Take the partial derivatives of 

x:

y:

The slope is 

To consider finding the slope, let's discuss the topic of the gradient.

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 rule will be necessary:

Derivative of an exponential: 

Note that u 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.

Take the partial derivatives of 

x:

y:

The slope is 

To consider finding the slope, let's discuss the topic of the gradient.

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

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.

Take the partial derivatives of 

x:

y:

z:

The slope is 

Use the product rule to find this derivative.

Use the product rule to find this derivative.

Recall that the derivative of a constant is zero.

Thus, the derivative is 

The points of inflection of a function occur where the second derivative of the funtion is equal to zero.

Find this second derivative by taking the derivative of the function twice:

Set the second derivative to zero and find the values that satisfy the equation:

Now, plug these values back in to the original function to find the values of the function that match to them:

The two points of inflection are

 can be shown to be to be a point of inflection by observing the sign change at lower and higher values 

The points of inflection of a function occur where the second derivative of the funtion is equal to zero.

Find this second derivative by taking the derivative of the function twice:

Set the second derivative to zero and find the value that satisfy the equation:

Now, plug this value back in to the original function to find the value of the function that matches:

The point of inflection is 

It can be confirmed that  is a point of inflection due to the sign change around this point. Picking a greater and lower value , observe the difference in sign of the second derivative:

The points of inflection of a function occur where the second derivative of the funtion is equal to zero.

Find this second derivative by taking the derivative of the function twice:

Set the second derivative to zero and find the values that satisfy the equation:

These can be shown to be points of inflection by plotting  and noting that it crosses the x-axis at these points; the sign of the function changes at them:

Now, plug these values back in to the original function to find the values of the function that match to them:

The points of inflection are

,, and 

Use the quotient rule to find this derivative.

Remember that the quotient rule is:

Apply this to our problem to get

Use the quotient rule to find this derivative.

Remember that the quotient rule is:

Apply this to our problem to get