Calculus Derivatives and Tangent Lines

This video tutorial covers the derivatives of logarithmic functions, including natural logs and logs with different bases. It explains the rules for finding derivatives of composite functions, and demonstrates the use of the chain rule, quotient rule, and product rule. The tutorial includes practice problems and solutions, showing different methods to find derivatives. It also covers finding the equation of a tangent line at a given point and verifying the derivative using a graphing calculator.

What is the derivative of the natural logarithm of x?

• 1/x

• ln(x)

• x

• e^x

Which formula is used to find the derivative of a logarithm with a base other than e?

• ln(x)/x

• 1/(ln(a) * x)

• e^x

• 1/x

What is the first step in proving the derivative of ln(x)?

• Differentiate directly

• Apply the quotient rule

• Use the chain rule

• Rewrite as an exponential equation

How can the expression ln(6x^2) be simplified using logarithmic properties?

• ln(6) - ln(x^2)

• ln(6x) + ln(x)

• ln(6) * ln(x^2)

• ln(6) + 2ln(x)

When applying the quotient rule, what is the first step?

• Differentiate the denominator

• Simplify the expression

• Differentiate the numerator

• Set up the quotient rule

In the product rule, what is the derivative of e^(2x) * ln(x)?

• 2e^(2x) * ln(x)

• e^(2x) + ln(x)

• e^(2x) * ln(x)

• 2e^(2x) * ln(x) + e^(2x)/x

What is the derivative of ln(x^2 - 5) - ln(x)?

• x^2 - 5/x

• 2x - 5/x

• 1/(x^2 - 5) - 1/x

• 2x/(x^2 - 5) - 1/x

What is the equation of the tangent line to the function at x = 0.5?

• y = -2.7726x + 1.8668

• y = 2.7726x + 0.4805

• y = -2.7726x - 0.4805

• y = 2.7726x - 1.8668

How can you verify the derivative value using a graphing calculator?

• Use the addition function

• Use the derivative function

• Use the subtraction function

• Use the multiplication function

What is the slope of the tangent line at the point of tangency?

• Approximately -2.7726

• Approximately 2.7726

• Approximately 1.8668

• Approximately 0.4805