Derivative of a Function Involving Logarithms

The video tutorial explains how to find the derivative of a function involving a logarithm. It begins by using the power property of logarithms to simplify the function. The product rule is then applied to differentiate the function, followed by calculating the derivatives of individual components. The expression is simplified, and the final derivative is presented. The tutorial concludes with a summary of the steps taken.

What is the initial function given in the problem?

• f(x) = 5x * log base 7 of x^2

• f(x) = 5x^2 * log base 7 of sqrt(x)

• f(x) = 5x^2 * log base 10 of sqrt(x)

• f(x) = 5x^2 * log base 7 of x

How is the square root of x expressed using a rational exponent?

• x^(1/2)

• x^2

• x^(2/3)

• x^1

What property of logarithms is used to simplify the function?

• Product property

• Quotient property

• Power property

• Change of base property

Which rule is applied to differentiate the product of two functions?

• Quotient rule

• Chain rule

• Product rule

• Power rule

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

• 1 / (a * ln(x))

• x / ln(a)

• ln(a) / x

• 1 / (ln(a) * x)

What is the derivative of 5/2 x^2 with respect to x?

• 5x

• 5/2 x

• 10x

• 5/2

What happens to the x in the denominator when simplifying the first product?

• It remains unchanged

• It cancels out completely

• It becomes x^2

• It simplifies to one factor of x

What is the final form of the derivative function?

• 5x^2 / (2 ln(7)) + 5 log base 7 of x

• 5x / (2 ln(7)) + 5x log base 7 of x

• 5x / (ln(7)) + 5x log base 7 of x

• 5 / (2 ln(7)) + x log base 7 of x

What is the base of the logarithm in the given function?

• 7

• 5

• 10

• e

What is the purpose of using the power property of logarithms in this problem?

• To convert the logarithm to a natural logarithm

• To eliminate the logarithm

• To change the base of the logarithm

• To simplify the differentiation process