Derivatives and Logarithmic Functions

The video tutorial covers the derivatives of exponential and logarithmic functions when the base is not equal to e. It explains the derivative rules, provides proofs for these formulas, and demonstrates their application in solving mathematical problems. The tutorial also covers the product and quotient rules and applies these concepts to a real-world scenario involving the depreciation of an office machine using the double declining balance method.

What is the derivative of an exponential function with base 'a'?

• x^a * ln(x)

• a^x / ln(a)

• a^x * ln(a)

• ln(a) / a^x

How do you find the derivative of log base 'a' of x?

• 1 / (x * ln(a))

• x / ln(a)

• ln(a) / x

• 1 / (a * ln(x))

Which property of logarithms is used to derive the derivative of a^x?

• Product property

• Power property

• Quotient property

• Change of base formula

In the example with base 5, what is the derivative of 5^(x^3 - 4)?

• 5^(x^3 - 4) / ln(5)

• 3x^2 * ln(5)

• 5^(x^3 - 4) * ln(5)

• ln(5) * 5^(x^3 - 4) * 3x^2

What is the derivative of log base 3 of (2x^2 - 5x)?

• 1 / (ln(3) * (2x^2 - 5x)) * (4x - 5)

• ln(3) / (2x^2 - 5x)

• 1 / (2x^2 - 5x)

• ln(3) * (4x - 5)

When applying the product rule, what is the derivative of log base 9 of x times 8^x?

• 1 / (ln(9) * x) + ln(8) * 8^x

• ln(8) * 8^x / ln(9)

• 8^x * ln(9)

• ln(9) * ln(8) * 8^x

How can natural log 9 be simplified using logarithmic properties?

• ln(9^2)

• ln(3^2)

• ln(2^3)

• ln(3) * ln(2)

What is the derivative of 2^x using the quotient rule?

• ln(2) * 2^x

• 2^x / ln(2)

• ln(2) / 2^x

• 2^x * ln(x)

In the depreciation example, what is the derivative of the salvage value function?

• ln(0.85) * 4600

• 4600 * 0.85^t

• 4600 / ln(0.85)

• 4600 * ln(0.85) * 0.85^t

What does a negative derivative indicate in the context of the office machine's value?

• The value is increasing

• The value is unpredictable

• The value is constant

• The value is decreasing