Calculating Derivatives and Exponents

The video tutorial explains how to find the derivative of the function f(x) = 3 - 7/x + 2/x^2 and evaluate it at x=2. It begins by rewriting terms with negative exponents to apply the power rule of differentiation. The tutorial then demonstrates the calculation of the derivative and simplifies it using positive exponents. Finally, it evaluates the derivative at x=2 to find the slope of the tangent line and verifies the result graphically.

What is the initial function given in the problem?

• f(x) = 3 + 7x - 2x^2

• f(x) = 3 + 7/x - 2/x^2

• f(x) = 3 - 7/x + 2/x^2

• f(x) = 3 - 7x + 2x^2

Why do we need to rewrite terms using negative exponents?

• To find the integral of the function

• To apply the power rule of differentiation

• To simplify the function

• To make the function continuous

What is the exponent property used to rewrite the function?

• a^-M = 1/a^-M

• a^M = 1/a^M

• 1/a^M = a^M

• 1/a^M = a^-M

How is the function rewritten using negative exponents?

• f(x) = 3 - 7x^-1 + 2x^-2

• f(x) = 3 + 7x^-1 - 2x^-2

• f(x) = 3 + 7x^1 - 2x^2

• f(x) = 3 - 7x^1 + 2x^2

What is the derivative of the constant term in the function?

• -3

• 0

• 1

• 3

What is the derivative of 7x^-1 using the power rule?

• 7x^2

• -7x^2

• 7x^-2

• -7x^-2

How is the derivative rewritten using positive exponents?

• 7/x^3 - 4/x^2

• 7x^-2 - 4x^-3

• 7/x^2 - 4/x^3

• 7x^2 - 4x^3

What is the value of f'(2)?

• 4/5

• 7/4

• 2/5

• 5/4

What is the common denominator used to simplify the expression for f'(2)?

• 4

• 2

• 8

• 16

What does f'(2) represent in the context of the function?

• The slope of the tangent line at x = 2

• The maximum value of the function

• The area under the curve at x = 2

• The y-intercept of the function