Differentiation of Radical Functions

The video tutorial explains how to find the derivative of a function with a variable under a radical sign using the power rule. It outlines three steps: rewriting the function with a fractional exponent, differentiating using the power rule, and simplifying the result. The tutorial provides a detailed example of differentiating f(x) = 5 times the cube root of x squared, demonstrating the conversion of the radical to a fractional exponent, applying the power rule, and simplifying the derivative to its simplest form using radical notation.

What is the first step in finding the derivative of a function with a variable under a radical sign?

• Rewrite the function with the variable as the base of an exponent.

• Apply the quotient rule.

• Use the chain rule.

• Directly differentiate the function.

Which rule is primarily used to differentiate a function with a single term under a radical sign?

• Quotient rule

• Power rule

• Chain rule

• Product rule

How do you express the cube root of x squared using exponents?

• x^1/3

• x^2/3

• x^3

• x^3/2

What is the equivalent expression for the nth root of a variable raised to the m power?

• a^(m+n)

• a^(m-n)

• a^(m/n)

• a^(n/m)

Why is it beneficial to convert a function from radical to exponential form before differentiating?

• It eliminates the need for simplification.

• It reduces the number of terms.

• It simplifies the application of the power rule.

• It makes integration easier.

What is the next step after rewriting the function in terms of exponents?

• Apply the chain rule.

• Convert back to radical form.

• Differentiate using the power rule.

• Simplify the expression.

According to the power rule, what is the derivative of x^n?

• n * x^(n+1)

• n * x^(n-1)

• x^(n-1)

• n * x^n

What is the derivative of 5x^(2/3) using the power rule?

• 5 * x^(1/3)

• 10/3 * x^(-1/3)

• 5/3 * x^(2/3)

• 10/3 * x^(1/3)

What is the exponent of x after applying the power rule to x^(2/3)?

• 1/3

• 0

• -1/3

• 2/3

What is the coefficient of x after differentiating 5x^(2/3) using the power rule?

• 10/3

• 2/3

• 5/3

• 5