Understanding Derivatives of Logarithmic Functions

This video tutorial explains how to determine the derivative of logarithmic functions when the base is not e. It covers two main derivative formulas: one without the chain rule and one with the chain rule. The video provides detailed examples, including the derivative of f(x) = 2 log base 5 of x without using the chain rule, and the derivative of g(x) = log base 2 of (x^4 + 3x) using the chain rule. The tutorial concludes with a summary of the key points discussed.

What is the derivative of log base a of x with respect to x without using the chain rule?

• a divided by (natural log x)

• 1 divided by (a times x)

• x divided by (natural log a)

• 1 divided by (natural log a times x)

In the function f(x) = 2 log base 5 of x, what is the value of 'a'?

• 2

• 5

• e

• 10

Why is the chain rule not needed for the function f(x) = 2 log base 5 of x?

• Because the base is e

• Because the input is just x

• Because the function is linear

• Because the input is a constant

What is the derivative of log base a of u with respect to x using the chain rule?

• u divided by (natural log a times x)

• 1 divided by (a times u)

• 1 divided by (natural log a times x)

• 1 divided by (natural log a times u times u')

In the function g(x) = log base 2 of (x^4 + 3x), what is the value of 'a'?

• 10

• 2

• 5

• e

What is the expression for u in the function g(x) = log base 2 of (x^4 + 3x)?

• x^4 + 3x

• x^3 + x

• x^3 + 3

• x^4 + 3

What is the derivative of u, denoted as u', for u = x^4 + 3x?

• 4x^3 + 3

• 3x^2 + 3

• 4x^2 + 3

• 4x^3 + x

What is the final expression for g'(x) in the function g(x) = log base 2 of (x^4 + 3x)?

• (4x^3 + 3) / (natural log e * (x^4 + 3x))

• (4x^3 + 3) / (natural log 5 * (x^4 + 3x))

• (4x^3 + 3) / (natural log 2 * (x^4 + 3x))

• (4x^3 + 3) / (natural log 10 * (x^4 + 3x))

What is the role of the chain rule in finding the derivative of g(x) = log base 2 of (x^4 + 3x)?

• It helps differentiate the exponent

• It helps differentiate the constant

• It helps differentiate the input function

• It helps differentiate the base

What is the importance of understanding derivatives of logarithmic functions?

• To solve linear equations

• To understand exponential growth

• To analyze logarithmic growth and decay

• To calculate integrals