Question

CALCULUS

what would be the derivative of this exponential function using the quotient rule

P= 1000 / 1+6e^-t


the e is not a variable if you know what im saying

Answer 1

`\displaystyle P' = (6000e^(t))/((6 + e^(t))^2)`

General Formulas and Concepts:

Algebra I

Functions

• Function Notation

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. 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))`

Exponential Derivatives

Explanation:

Step 1: Define

Identify

`\displaystyle P = (1000)/(1 + 6e^(-t))`

Step 2: Differentiate

1. Derivative Rule [Quotient Rule]:
   `\displaystyle P' = ((1000)'(1 + 6e^(-t)) - 1000(1 + 6e^(-t))')/((1 + 6e^(-t))^2)`
2. Basic Power Rule:
   `\displaystyle P' = (0(1 + 6e^(-t)) - 1000(1 + 6e^(-t))')/((1 + 6e^(-t))^2)`
3. Simplify:
   `\displaystyle P' = ( -1000(1 + 6e^(-t))')/((1 + 6e^(-t))^2)`
4. Rewrite [Derivative Property - Addition/Subtraction]:
   `\displaystyle P' = ( -1000 \bigg[ (1)' + (6e^(-t))' \bigg] )/((1 + 6e^(-t))^2)`
5. Basic Power Rule:
   `\displaystyle P' = ( -1000 \bigg[ 0 + (6e^(-t))' \bigg] )/((1 + 6e^(-t))^2)`
6. Rewrite [Derivative Property - Multiplied Constant]:
   `\displaystyle P' = ( -1000 \bigg[ 0 + 6(e^(-t))' \bigg] )/((1 + 6e^(-t))^2)`
7. Exponential Derivative:
   `\displaystyle P' = ( -1000 \bigg[ 0 + -6e^(-t) \bigg] )/((1 + 6e^(-t))^2)`
8. Simplify:
   `\displaystyle P' = (6000e^(-t))/((1 + 6e^(-t))^2)`
9. Rewrite:
   `\displaystyle P' = (6000e^(t))/((6 + e^(t))^2)`

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

Unit: Derivatives

Book: College Calculus 10e