Derivatives of Logarithmic and Exponential Functions

The video tutorial covers advanced differentiation techniques, focusing on logarithmic and exponential functions. It explains the rules for differentiating these functions, including the use of the chain rule for composite functions. The tutorial also provides a proof for the derivative of e^x using series expansion, highlighting the unique property of e^x being equal to its derivative. The content is designed to enhance understanding of calculus concepts and prepare students for more complex applications.

What is the derivative of the natural logarithm of x?

• ln(x)

• 1/x

• x

• e^x

How do you express the derivative of log base a of x using the change-of-base formula?

• x ln(a)

• a^x

• ln(x)/ln(a)

• 1/(x ln(a))

When differentiating the natural log of sine x, what rule is primarily used?

• Chain Rule

• Power Rule

• Quotient Rule

• Product Rule

What is the derivative of log base 10 of x squared?

• 2/ln(10)x

• x^2/ln(10)

• 2x/ln(10)

• 1/ln(10)x^2

What is the derivative of an exponential function a^x?

• a^x ln(x)

• x a^x

• ln(a) a^x

• a^x

How do you differentiate 2 raised to the power of 3x + 1?

• ln(3) 2^(3x+1)

• 2^(3x+1)

• 3 ln(2) 2^(3x+1)

• ln(2) 2^(3x+1)

Why is the derivative of e^x equal to e^x?

• Because e is a constant

• Due to the power rule

• Because the series expansion of e^x remains unchanged

• Because e^x is a linear function