Understanding Derivatives and the Product Rule

The video tutorial explains how to determine the derivative of a function using the product rule. It begins by identifying the function as a product of two functions and rewriting terms with rational exponents. The product rule is applied to find the derivative, followed by algebraic simplification to combine like terms. The final derivative is presented in both rational and radical forms, with additional tips on rewriting expressions for clarity.

What is the product rule used for in calculus?

• Finding the derivative of a sum of functions

• Finding the limit of a function

• Finding the derivative of a product of two functions

• Finding the integral of a function

How can the square root of x be rewritten to apply the product rule?

• As x to the power of -1/2

• As x to the power of 2

• As x to the power of 1/2

• As x to the power of 3/2

What is the derivative of 3x?

• 1

• x

• 0

• 3

What is the derivative of x to the 1/2?

• 1/2x to the 1/2

• 1/2x to the -1/2

• x to the 1/2

• x to the -1/2

When simplifying the derivative expression, what operation is performed on the exponents?

• Multiplication

• Subtraction

• Addition

• Division

What is the result of multiplying 2x squared by 1/2x to the -1/2?

• x to the 3/2

• x to the 1/2

• x cubed

• x squared

What is the simplified form of -5/2x to the -1/2?

• -5/2 square root x

• -5/2x to the 1/2

• -5/2x squared

• -5/2x cubed

How can x to the 3/2 be rewritten in radical form?

• x cubed

• x squared

• x cube root x

• x square root x

What is the final expression for the derivative function?

• 18x squared + 5x to the 1/2 - 5/2 square root x - 15

• 18x squared + 5x to the 3/2 - 5/2 square root x + 15

• 18x squared + 5x to the 3/2 - 5/2 square root x - 15

• 18x squared + 5x to the 3/2 + 5/2 square root x - 15

Why might some textbooks rewrite x to the 3/2 as x square root x?

• To simplify the expression

• To make it more complex

• To match the original function

• To avoid using radicals