Understanding Derivatives and Exponents

The video tutorial explains how to find the derivative of a function, f(x), and evaluate it at a specific point, x=2. It covers the use of power rules for differentiation, handling rational and negative exponents, and simplifying expressions. The tutorial also demonstrates how to calculate decimal approximations and verify results using a calculator. The process is illustrated with examples, including rewriting functions and applying differentiation rules.

What is the derivative of 6e^x?

• 6x^3

• 6x^2

• 6e^x

• 6

What is the derivative of a constant, such as 24?

• x

• 24

• 0

• 1

How do you find f'(2) for the function f'(x) = 6e^x - 12x^2?

• Substitute x = 1

• Substitute x = -2

• Substitute x = 2

• Substitute x = 0

What is the rational exponent form of the fourth root of x?

• x^4

• x^1/2

• x^2

• x^1/4

How is 3 divided by x^8 expressed using negative exponents?

• 3x^8

• 3x^2

• 3x^-8

• 3/x^8

What is the simplified form of the derivative of 6x^(1/4)?

• 3/2 x^(-3/4)

• 6x^(1/4)

• 3x^(1/4)

• 6x^(-1/4)

How can x^(-3/4) be rewritten using positive exponents?

• x^(1/4)

• x^(-3/4)

• x^(3/4)

• 1/x^(3/4)

What is the decimal approximation of f'(2) for the function 6x^(1/4) - 3x^(-8)?

• 1.2345

• 0.9388

• 1.6789

• 0.5678

Which calculator function is used to verify the derivative numerically?

• Option 8

• Option 5

• Option 1

• Option 3

Why might there be a slight difference in numerical derivative values on a calculator?

• Due to rounding errors

• Due to battery issues

• Because of the algorithm used

• Because of incorrect input