Economics at Zero Marginal Cost: Exponential & Logarithmic Rules

Wednesday, June 10, 2020

Exponential & Logarithmic Rules

Rues of Differentiation

1. Exponential Rule:

Rules of differentiation for exponential functions:

1. If $y=e^{x}$

$\frac{d}{dx}e^x=e^x$ or ${f}'(x)=e^x$

2. If $y=e^a^x$

$\frac{d}{dx}e^a^x=ae^a^x$ or $f'(x)=ae^a^x$

3. If $y=e^{-x}$

$\frac{d}{dx}e^-^x=-e^-^x$ or $f'(x)=-e^-^x$

Excercise:

i) If $y=e^x$

$\\\frac{dy}{dx}=\frac{d}{dx}y=\frac{d}{dx}e^x\\ \frac{dy}{dx}=e^x$

ii) If $y=e^4^x\\$

$\\\frac{dy}{dx}=\frac{d}{dx}y=\frac{d}{dx}e^4^x\\ \frac{dy}{dx}=4e^4^x$

iii) If $y=e^-^x$

$\\\frac{dy}{dx}=\frac{d}{dx}y=\frac{d}{dx}e^-^x\\ \frac{dy}{dx}=-e^-^x$

iv) If $y=e^{-7x}$

$\\\frac{dy}{dx}=\frac{d}{dx}y=\frac{d}{dx}e^{-7x}\\ \frac{dy}{dx}=-7e^{-7x}$

2. Logarithmic Rule:

The derivative of a logarithmic function $y=f(x)=log\;x$ is $\frac{1}{x}$

Symbolically,

$\frac{d}{dx}log\;x=\frac{1}{x}$ or $f'(x)=\frac{1}{x}$

Excercise

i) If $y= log\;x$

SOLUTION

$\\\frac{dy}{dx}=\frac{d}{dx}y=\frac{d}{dx}\;log\;x\\ \frac{dy}{dx}=\frac{1}{x}$

Click here for Lecture Notes

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)