1. ## y=4e^x

$y=4e^x$
How to find $\frac{dy}{dx}$ ?

Method I:
$y=4e^x$ ...... 1
$y+\delta y=4e^{x+\delta x}$ ...... 2

2-1
$\delta y= 4e^{x+\delta x}-4e^x$
$\delta y= 4e^x\cdot e^{\delta x}-4e^x$
$\delta y= 4e^x(e^{\delta x}-1)$
$\frac{\delta y}{\delta x}= \frac{4e^x(e^{\delta x}-1)}{\delta x}$
$\frac{\delta y}{\delta x}= 4e^x\cdot \frac{(e^{\delta x}-1)}{\delta x}$
$\lim_{\delta x \rightarrow 0}{\frac{\delta y}{\delta x}}= 4e^x\cdot \frac{(1-1)}{0}$
$\lim_{\delta x \rightarrow 0}{\frac{\delta y}{\delta x}}= 4e^x\cdot \frac{(0)}{0}$
$\lim_{\delta x \rightarrow 0}{\frac{\delta y}{\delta x}}= 4e^x$

Method II:
$y= 4e^x$
$\frac{y}{4}=e^x$
$\ln\left(\frac{y}{4}\right)=x$
$\frac{4}{y}=\frac{dx}{dy}$
$\frac{dy}{dx}=\frac{y}{4}$
$\frac{dy}{dx}=e^x$

Which method is true?
I think method II is wrong and I am not sure the 0/0 fraction in method I is true or not...because 0 as denominator is undefined...
Anyone can help me?

2. Originally Posted by SengNee
$y=4e^x$
How to find $\frac{dy}{dx}$ ?

Method I:
$y=4e^x$ ...... 1
$y+\delta y=4e^{x+\delta x}$ ...... 2

2-1
$\delta y= 4e^{x+\delta x}-4e^x$
$\delta y= 4e^x\cdot e^{\delta x}-4e^x$
$\delta y= 4e^x(e^{\delta x}-1)$
$\frac{\delta y}{\delta x}= \frac{4e^x(e^{\delta x}-1)}{\delta x}$
$\frac{\delta y}{\delta x}= 4e^x\cdot \frac{(e^{\delta x}-1)}{\delta x}$
$\lim_{\delta x \rightarrow 0}{\frac{\delta y}{\delta x}}= 4e^x\cdot \frac{(1-1)}{0}$
$\lim_{\delta x \rightarrow 0}{\frac{\delta y}{\delta x}}= 4e^x\cdot \frac{(0)}{0}$
$\lim_{\delta x \rightarrow 0}{\frac{\delta y}{\delta x}}= 4e^x$

Method II:
$y= 4e^x$
$\frac{y}{4}=e^x$
$\ln\left(\frac{y}{4}\right)=x$
$\frac{4}{y}=\frac{dx}{dy}$
$\frac{dy}{dx}=\frac{y}{4}$
$\frac{dy}{dx}=e^x$

Which method is true?
I think method II is wrong and I am not sure the 0/0 fraction in method I is true or not...because 0 as denominator is undefined...
Anyone can help me?
yes, method II is wrong, as you ended up with the wrong answer, but, in any case, you are making things wwaaayyyy too complicated.

recall that $\frac d{dx}cf(x) = c \frac d{dx}f(x)$

where $c$ is a constant.

in other words, when finding the derivative of a function with a constant in front of it, we can simply find the derivative of the function without the constant, and then multiply it by the constant.

so, $y = 4 e^x$

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

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

the general rule: $\frac d{dx}ce^x = ce^x$ for $c$ some constant, so you could actually omit the second line altogether