Understanding Derivatives and Limits

This video tutorial covers the process of finding derivatives using both the power rule and the limit process. It begins with an introduction to derivatives and the power rule, followed by examples of applying the power rule to various functions. The tutorial then explores the derivatives of constants and simple functions. It delves into using limits to find derivatives, including rational and square root functions, and concludes with advanced techniques for evaluating limits.

What is the derivative of a constant function?

• One

• Zero

• The constant times x

• The constant itself

Using the power rule, what is the derivative of x^5?

• 4x^5

• 5x^5

• x^4

• 5x^4

What is the limit definition of a derivative?

• The limit as h approaches zero of [f(x+h) - f(x)]/h

• The slope of the tangent line

• The rate of change of a function

• The integral of a function

How do you find the derivative of x^2 using limits?

• By applying the power rule

• By differentiating twice

• By using the limit as h approaches zero of [f(x+h) - f(x)]/h

• By integrating the function

What is the derivative of x^2 + 5x - 7 using limits?

• 2x - 7

• 5x - 7

• x^2 + 5

• 2x + 5

What is the derivative of 1/x using the power rule?

• -1/x^2

• x^2

• 1/x^2

• -x^2

How do you simplify a limit involving radicals?

• By integrating

• By multiplying by the conjugate

• By factoring

• By using the power rule

What is the derivative of the square root of x using limits?

• 1/(2√x)

• √x

• 2√x

• 1/√x

How can you evaluate the derivative at a specific point using limits?

• By using the power rule

• By substituting the point into the derivative function

• By using the limit as x approaches a of [f(x) - f(a)]/(x-a)

• By integrating the function

What is the derivative of x^4 using the power rule?

• 4x^4

• 4x^3

• x^3

• 3x^4