Understanding Derivatives of Exponential Functions

The video tutorial explains how to take the derivative of 2 to the x power by rewriting it in terms of e using natural logarithms. It demonstrates the use of exponent properties and the chain rule to simplify the derivative. The final result shows that the derivative of 2 to the x is the natural log of 2 times 2 to the x.

What is the main challenge in finding the derivative of 2 to the x power?

• The base is not e.

• The exponent is not a constant.

• The base is a negative number.

• The function is not continuous.

How can 2 be expressed in terms of e?

• 2 times e

• e to the power of 2

• e to the power of the natural log of 2

• 2 to the power of e

What exponent property is used to simplify the expression?

• a to the b times c is a to the b plus c

• a to the b raised to the c is a to the b times c

• a to the b raised to the c is a to the b plus c

• a to the b times c is a to the b times a to the c

What rule is applied to differentiate e to the natural log of 2 times x?

• Chain Rule

• Power Rule

• Quotient Rule

• Product Rule

What is the derivative of e to the something with respect to that something?

• Natural log of something

• 0

• 1

• e to the something

What is the derivative of a constant times x with respect to x?

• The constant times x

• x

• Zero

• The constant

What is the final simplified form of the derivative of 2 to the x?

• 2 to the x times the natural log of 2

• e to the x times 2

• 2 to the x times e

• e to the x times the natural log of 2

Why is e to the natural log of 2 equal to 2?

• Because natural log of 2 is 2

• Because natural log of 2 is 1

• Because e is a constant

• Because e raised to the natural log of a number is the number itself

What is the purpose of rewriting 2 to the x as e to the natural log of 2 times x?

• To make it easier to integrate

• To apply the chain rule for differentiation

• To convert it into a polynomial

• To simplify the base

What is the role of the natural log of 2 in the derivative of 2 to the x?

• It is the derivative of x

• It is the base of the exponent

• It acts as a constant multiplier

• It is the exponent