Calculus Derivatives and Exponents

The video tutorial explains how to find the derivative of functions using rational exponents and the power rule. It covers the process of rewriting square roots as rational exponents, applying the power rule, and simplifying the results. The tutorial demonstrates these steps with two functions, g(x) and f(x), and emphasizes expressing derivatives with positive exponents. The video concludes with a brief encouragement to the viewer.

What is the first step in finding the derivative of a function involving a square root?

• Use the chain rule

• Rewrite the square root using a rational exponent

• Apply the product rule

• Differentiate directly

Which rule is used to find the derivative of g(x) = 4√x - sin(x)?

• Quotient rule

• Product rule

• Power rule

• Chain rule

What is the derivative of sin(x) with respect to x?

• sin(x)

• -sin(x)

• cos(x)

• -cos(x)

How can the expression 2x^(-1/2) be rewritten using positive exponents?

• 2/x^2

• 2x^(1/2)

• 2/x^(1/2)

• 2√x

What is another way to express x^(1/2)?

• √x

• x^2

• 1/x^2

• 1/√x

What is the final expression for the derivative of g(x) in radical form?

• 2x - cos(x)

• 2/x - sin(x)

• 2√x + cos(x)

• 2/√x - cos(x)

What is the initial form of the function f(x) before finding its derivative?

• 5x^3 + cos(x)

• 5x^(-3) - sin(x)

• -5x^(-3) - cos(x)

• -5x^3 + sin(x)

Tags

CCSS.6.EE.A.1

What is the derivative of cos(x) with respect to x?

• -sin(x)

• cos(x)

• sin(x)

• -cos(x)

Tags

CCSS.6.EE.A.1

How is the expression 15x^(-4) rewritten using positive exponents?

• 15√x

• 15/x^(1/4)

• 15/x^4

• 15x^4

What is the simplified form of the derivative of f(x) = -5x^(-3) - cos(x)?

• 15/x^4 + sin(x)

• 15x^4 + sin(x)

• 15x^(-4) - sin(x)

• 15/x^4 - sin(x)