Understanding the Derivative of a^x

The video tutorial provides a detailed proof of the derivative of a^x with respect to x, showing that it equals the natural log of a times a^x. It begins by introducing the concept and using logarithmic definitions to express the equation. The tutorial then applies implicit differentiation and the chain rule, followed by the change of base formula for logarithms. Finally, it calculates the derivative and concludes with a summary of the proof.

What is the initial step in proving the derivative of a^x?

• Using the power rule

• Using integration by parts

• Setting up a logarithmic equation

• Applying the product rule

What is the equivalent logarithmic equation for a^x = y?

• log base x of a = y

• log base a of y = x

• log base y of a = x

• log base a of x = y

Why is implicit differentiation used in this proof?

• To avoid using the chain rule

• To handle the term containing y

• To directly find the derivative

• To simplify the equation

How is the derivative of log base a of y with respect to x expressed?

• x times dx/dy

• 1 over y times dy/dx

• 1 over x times dx/dy

• y times dy/dx

What mathematical rule is applied due to the presence of y in the equation?

• Product rule

• Quotient rule

• Chain rule

• Power rule

What is the purpose of the change of base formula in this proof?

• To find the integral of the function

• To eliminate the need for implicit differentiation

• To convert the base to 10

• To simplify the differentiation process

What is the role of the natural log in the change of base formula?

• It is used to find the integral

• It is irrelevant to the proof

• It serves as a constant

• It acts as a variable

What is the derivative of x with respect to x?

• x^2

• x

• 1

• 0

What is the final expression for the derivative of a^x?

• a^x * x

• x^a * natural log a

• a^x * natural log a

• a^x * log base a

What is the final step in solving for dy/dx in the proof?

• Dividing both sides by y

• Multiplying both sides by y and natural log a

• Subtracting y from both sides

• Adding natural log a to both sides