Differentiating Logarithmic and Exponential Functions

Professor Dave explains how to differentiate logarithmic and exponential functions. He covers the rules for differentiating natural logs and logs with different bases, using the chain rule for composite functions. The video also explores differentiating exponential functions, including e^x, and provides a proof using series. The session concludes with a discussion on the graphical implications of these derivatives.

What is the derivative of the natural log of x?

• 1/x

• x/ln(x)

• ln(x)

• x

How is the derivative of a logarithm with base a expressed?

• 1/(ln(a) * x)

• ln(x)/ln(a)

• 1/ln(a)

• ln(a)/x

What is the result of differentiating the natural log of sine x using the chain rule?

• cos(x)/sin(x)

• cot(x)

• sin(x)/cos(x)

• 1/sin(x)

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

• 1/(ln(10) * x^2)

• 2x/ln(10)

• ln(10)/x^2

• 2/(ln(10) * x)

What is the derivative of 2 to the power of x?

• 2x

• ln(2) * 2^x

• 2^x

• x * ln(2)

How do you differentiate an exponential function where the exponent is a function of x?

• A^(derivative of the exponent)

• ln(A) * A^x * derivative of the exponent

• A^x * ln(A)

• ln(A) * derivative of the exponent

What is the derivative of e to the power of x?

• 1/e

• ln(e) * e^x

• x * e^x

• e^x

Using the series expansion, what happens to the series representing e^x when differentiated?

• It remains unchanged

• It doubles

• It halves

• It becomes zero

What does it imply if the function and its derivative are graphically the same?

• Each point on the function is equal to its slope

• The function is non-differentiable

• The function is linear

• The function has no slope

What fundamental property of e^x is demonstrated through its series representation?

• It is equal to its derivative

• It is a constant function

• It is an exponential growth function

• It has a variable slope