Mathematics, 18.07.2019 12:30 sierrawalk3896

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

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Answer from: jedsgorditas

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

General Formulas and Concepts:

Algebra I

Functions

Function Notation

Calculus

Differentiation

DerivativesDerivative 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:

f(x) = cxⁿf’(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

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

Answer from: Quest

1,589.63 in.^3

step-by-step explanation:

i hope this you : d

Answer from: Quest

step-by-step explanation:

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