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

Answer:

$\displaystyle P' = \frac{6000e^{t}}{(6 + e^{t})^2}$

General Formulas and Concepts:

Algebra I

Functions

• Function Notation

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]:                                                                           $\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Exponential Derivatives

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle P = \frac{1000}{1 + 6e^{-t}}$

Step 2: Differentiate

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

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

Unit: Derivatives

Book: College Calculus 10e