Logarithmic and Exponential Derivatives

This lesson covers how to find derivatives of exponential and logarithmic functions that are not base e, using the chain rule. It includes detailed examples of finding the derivative of f(x) = 2 * 7^(3x^2 + 2) and G(x) = -3 log base 2 (12 - 5x). The lesson emphasizes the importance of identifying composite functions and applying the chain rule correctly, with step-by-step explanations and simplifications.

What is the primary rule used when finding derivatives of exponential and logarithmic functions that are not base e?

• Product Rule

• Chain Rule

• Power Rule

• Quotient Rule

In the function f(x) = 2 * 7^(3x^2 + 2), what is the base of the exponential function?

• 2

• 3

• 7

• e

What is the derivative of the exponent 3x^2 + 2 in the function f(x) = 2 * 7^(3x^2 + 2)?

• 6x

• 3x

• 2x

• 9x

After applying the chain rule, what is the simplified form of the derivative of f(x) = 2 * 7^(3x^2 + 2)?

• 6x * ln(7) * 7^(3x^2 + 2)

• 12x * ln(7) * 7^(3x^2 + 2)

• 12x * ln(2) * 7^(3x^2 + 2)

• 6x * ln(2) * 7^(3x^2 + 2)

Why is it important to identify U and U' when using the chain rule?

• To find the integral

• To correctly apply the chain rule

• To apply the product rule

• To simplify the function

What happens to the derivative of a function if U is equal to x?

• U' becomes 0

• U' becomes 1

• U' becomes x

• U' becomes x^2

In the function G(x) = -3 log base 2 of (12 - 5x), what is the base of the logarithm?

• 2

• 10

• 5

• e

What is the derivative of the input 12 - 5x in the function G(x) = -3 log base 2 of (12 - 5x)?

• -5

• -12

• 5

• 12

What is the simplified form of the derivative of G(x) = -3 log base 2 of (12 - 5x)?

• 15 / (ln(5) * (12 - 5x))

• -15 / (ln(2) * (12 - 5x))

• 15 / (ln(2) * (12 - 5x))

• -15 / (ln(5) * (12 - 5x))

What is the role of the natural logarithm in the derivative of exponential functions with a base other than e?

• It is used to divide the derivative

• It is used to add to the derivative

• It is used to multiply the derivative

• It is not used