Derivatives of Exponential and Logarithmic Functions

This lesson covers the derivatives of exponential and logarithmic functions, both with base e and other bases, without using the chain rule. It includes step-by-step calculations for functions like f(x) = 2 * e^x, g(x) = 2 * 5^x, h(x) = -5 * ln(x), and p(x) = -5 * log base 7 of x, explaining the derivative formulas and their applications.

What is the main focus of this lesson?

• Finding derivative functions of exponential and logarithmic functions

• Understanding the chain rule

• Finding integrals of exponential functions

• Solving differential equations

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

• x * e^x

• e^x

• e^x * ln(x)

• x^e

For the function f(x) = 2 * e^x, what is the derivative f'(x)?

• 2 * x * e^x

• 2 * ln(x)

• e^x

• 2 * e^x

What is the base of the exponential function G(x) = 2 * 5^x?

• 10

• 5

• 2

• e

What is the derivative of a^x with respect to x, where a is a constant?

• a^x * x

• a^x * ln(a)

• x * a^x

• a^x / ln(a)

What is the derivative of ln(x) with respect to x?

• 1/x

• ln(x)/x

• x * ln(x)

• x

For the function H(x) = -5 * ln(x), what is the derivative H'(x)?

• 5 * ln(x)

• 5 / x

• -5 / x

• -5 * x

What is the base of the logarithmic function P(x) = -5 * log base 7 of x?

• 10

• 5

• e

• 7

What is the derivative of log base a of x with respect to x?

• a^x / x

• x / ln(a)

• ln(a) / x

• 1 / (x * ln(a))

For the function P(x) = -5 * log base 7 of x, what is the derivative P'(x)?

• 5 * ln(7)

• -5 / (x * ln(7))

• 5 / (x * ln(7))

• -5 * ln(7)