Understanding Derivatives and the Product Rule

The video tutorial explains how to find the derivative of the function e^x times cosine of x using the product rule. It begins with a brief introduction to derivatives, followed by a discussion on the derivatives of e^x and cosine functions. The product rule is then introduced and explained in detail, with a step-by-step application to the given function. Finally, the expression is simplified, demonstrating the practical use of the product rule in calculus.

What is the derivative of e^x with respect to x?

• 1

• x^e

• e^x

• 0

What is the derivative of cosine of x with respect to x?

• negative cosine of x

• cosine of x

• sine of x

• negative sine of x

Which rule is used to find the derivative of a product of two functions?

• Product Rule

• Power Rule

• Chain Rule

• Quotient Rule

In the product rule, if u(x) = e^x and v(x) = cosine of x, what is u'(x)?

• e^x

• x^e

• negative sine of x

• cosine of x

In the product rule, if u(x) = e^x and v(x) = cosine of x, what is v'(x)?

• cosine of x

• sine of x

• e^x

• negative sine of x

What is the result of applying the product rule to e^x times cosine of x?

• e^x times sine of x

• e^x times cosine of x minus e^x times sine of x

• e^x plus cosine of x

• e^x minus cosine of x

What can be factored out from the expression e^x times cosine of x minus e^x times sine of x?

• x

• sine of x

• cosine of x

• e^x

What is the simplified form of the derivative of e^x times cosine of x?

• e^x times (cosine of x - sine of x)

• e^x times (cosine of x + sine of x)

• e^x times (sine of x + cosine of x)

• e^x times (sine of x - cosine of x)

What is the significance of the product rule in calculus?

• It helps in integrating functions.

• It simplifies trigonometric identities.

• It is used to find the derivative of a quotient.

• It allows differentiation of products of functions.

What is a key feature of the function e^x in calculus?

• It is always negative.

• Its derivative is e^x itself.

• Its derivative is zero.

• It is a constant function.