Differentiation Techniques and Rules

The video tutorial reviews the product rule of differentiation and its combination with the chain rule. It explains the product rule, which states that the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first. The tutorial applies this rule to find the derivative of a specific function, e^(3x) * sin(x^2), by identifying the components and using the chain rule to find their derivatives. The final result is simplified by factoring out common terms.

What does the product rule of differentiation state?

• The derivative of a quotient of two functions is the quotient of their derivatives.

• The derivative of a product of two functions is U * V' + V * U'.

• The derivative of a product of two functions is the product of their derivatives.

• The derivative of a sum of functions is the sum of their derivatives.

In the function FX = e^(3x) * sin(x^2), what is the derivative of U, where U = e^(3x)?

• sin(x^2)

• 3x * e^(3x)

• e^(3x)

• 3 * e^(3x)

What is the derivative of V, where V = sin(x^2), using the chain rule?

• sin(x^2)

• cos(x^2)

• 2x * cos(x^2)

• 2 * sin(x^2)

How do you apply the product rule to find the derivative of FX = e^(3x) * sin(x^2)?

• Multiply U and V directly.

• Use U * V' + V * U'.

• Add the derivatives of U and V.

• Subtract the derivative of V from U.

What is the final step in simplifying the derivative of FX = e^(3x) * sin(x^2)?

• Add e^(3x) to the result.

• Divide by e^(3x).

• Factor out e^(3x).

• Multiply by sin(x^2).