# Calculus Examples

,

Step 1

Assume .

Step 2

Step 2.1

Substitute values into .

Step 2.1.1

Substitute for .

Step 2.1.2

Substitute for .

Step 2.2

Since there is no log with negative or zero argument, no even radical with zero or negative radicand, and no fraction with zero in the denominator, the function is continuous on an open interval around the value of .

Continuous

Continuous

Step 3

Step 3.1

Set up the partial derivative.

Step 3.2

Since is constant with respect to , the derivative of with respect to is .

Step 3.3

Differentiate using the Power Rule which states that is where .

Step 3.4

Multiply by .

Step 4

Step 4.1

Substitute for .

Step 4.2

Since there is no log with negative or zero argument, no even radical with zero or negative radicand, and no fraction with zero in the denominator, the function is continuous on an open interval around the value of .

Continuous

Continuous

Step 5

Both the function and its partial derivative with respect to are continuous on an open interval around the value of .

One unique solution