Question

Find the derivative of f(x) =( -7/x) at x = -3.

pls help !!

Answer 1

`\displaystyle f'(-3) = (7)/(9)`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Calculus

Derivatives

Derivative Notation

Derivative of a constant is a 0

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]:
`\displaystyle (d)/(dx) [(f(x))/(g(x)) ]=(g(x)f'(x)-g'(x)f(x))/(g^2(x))`

Explanation:

Step 1: Define

`\displaystyle f(x) = (-7)/(x)`

`\displaystyle x = -3`

Step 2: Differentiate

1. Quotient Rule:
   `\displaystyle f'(x) = ((d)/(dx)[7]x - (d)/(dx)[x](-7))/(x^2)`
2. Basic Power Rule:
   `\displaystyle f'(x) = ((0)x - x^(1 - 1)(-7))/(x^2)`
3. Simplify:
   `\displaystyle f'(x) = ((0)x - x^0(-7))/(x^2)`
4. Simplify:
   `\displaystyle f'(x) = ((0)x - 1(-7))/(x^2)`
5. Multiply:
   `\displaystyle f'(x) = (0 + 7)/(x^2)`
6. Add:
   `\displaystyle f'(x) = (7)/(x^2)`

Step 3: Solve

1. Substitute in x [Derivative]:
   `\displaystyle f'(-3) = (7)/((-3)^2)`
2. Evaluate exponents:
   `\displaystyle f'(-3) = (7)/(9)`

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Derivatives

Book: College Calculus 10e