Understanding Derivatives and the Product Rule

The video tutorial explains how to find the derivative of a given function using the product rule. It begins by rewriting the function for easier differentiation, then applies the product rule to find the derivatives of each term. The tutorial continues with simplifying the expression and concludes with the final derivative result. The process involves understanding the product rule, calculating derivatives, and simplifying expressions.

What is the initial function given for differentiation?

• f(x) = x / 8 * ln(x) - x^8 / 64

• f(x) = x^8 * ln(x) - x / 64

• f(x) = x / 8 + ln(x) - x^8 / 64

• f(x) = x^8 / 8 * ln(x) - x / 64

Which rule is applied to differentiate the product of two functions?

• Quotient Rule

• Chain Rule

• Power Rule

• Product Rule

In the product rule, what is the derivative of U * V with respect to X?

• U * V + U' * V'

• U' * V' + U * V

• U' * V + U * V'

• U * V' + V * U'

What is the derivative of ln(x) with respect to x?

• x

• ln(x)

• 1/x

• 0

What is the derivative of 1/8 * x^8 with respect to x?

• 1/8 * x^8

• 8 * x^7

• 1/8 * 8 * x^7

• x^7

What is the derivative of 164 * x^8 with respect to x?

• 164 * x^7

• 8 * x^7

• 164 * 8 * x^7

• 164 * x^8

How does x^8 divided by x simplify?

• 1

• x

• x^8

• x^7

What common factor is shared by 8 and 64?

• 16

• 2

• 8

• 4

What is the final derivative function obtained?

• f'(x) = x^8 * ln(x)

• f'(x) = x^7

• f'(x) = x^7 * ln(x)

• f'(x) = ln(x)

What happens to the first and last terms in the simplification process?

• They cancel each other out.

• They add up.

• They multiply.

• They remain unchanged.