d/dx x^4x
is there a proof?

Question

amistre64 · Answer

a proof for what?

amistre64 · Answer

even if its ^(4x) the power rule and chain rules apply

anonymous · Answer

d/dx a^x=a^xlna

anonymous · Answer

Yes just apply the chain rule.

anonymous · Answer

can you work it out for me?

anonymous · Answer

$$(f(g(x))' = f'(g(x))g'(x)$$

amistre64 · Answer

f(x) = x^(g(x))
g(x) = 4(h(x))
h(x) = x

anonymous · Answer

huh how come it becomes a function...?

anonymous · Answer

composite function..

amistre64 · Answer

just by the nature of what a function is

amistre64 · Answer

you can rename the parts of an expression/equation to match seperate function

amistre64 · Answer

sqrt(5x) for example is:

sqrt(...)
5(...)
x

amistre64 · Answer

it helps to see the chain rule in action

anonymous · Answer

i cant see a chain rule..

anonymous · Answer

whats chain rule

zarkon · Answer

you want this ...

$$\frac{d}{dx}x^{4x}$$

amistre64 · Answer

f(x) = x^(g(x))
f'(x) = g(x) x^(g(x)-1)

g(x) = 4(h(x))
g'(x) = 4(h'(x))

h(x) = x
h'(x) = 1

zarkon · Answer

that is not true

amistre64 · Answer

the chain rule is akin to peeling an onion ... and deriving each section

zarkon · Answer

this ....

f(x) = x^(g(x)) 
f'(x) = g(x) x^(g(x)-1)

is not true

anonymous · Answer

then how to solve?

amistre64 · Answer

zarkon · Answer

$$\frac{d}{dx}x^{4x}=\frac{d}{dx}e^{4x\ln(x)}$$

zarkon · Answer

now use chain and product rule

anonymous · Answer

Maybe he/she want definition of derivative

amistre64 · Answer

im open to being mistaken at times ;)

anonymous · Answer

zarkon how did you arrive at that step

zarkon · Answer

$$\frac{d}{dx}x^{4x}=\frac{d}{dx}e^{4x\ln(x)}$$

$$=e^{4x\ln(x)}\left(\frac{4x}{x}+4\ln(x)\right)$$
$$=x^{4x}\left(4+4\ln(x)\right)$$

anonymous · Answer

how did the first step come about

zarkon · Answer

$$y=x^{4x}$$

$$\ln(y)=\ln(x^{4x})$$


$$\ln(y)=4x\ln(x)$$

$$y=e^{4x\ln(x)}$$

zarkon · Answer

ok?

anonymous · Answer

OK! thanks=)

anonymous · Answer

so whenever you see that you ln both sides?

zarkon · Answer

$$\frac{d}{dx}h(x)^{g(x)}$$

do this 

$$\frac{d}{dx}e^{g(x)\ln(h(x))}$$

zarkon · Answer

just never use the power rule.

that only applies when the exponent is a constant with respect to the variable you are differentiating with.

anonymous · Answer

is that memory work?

zarkon · Answer

?

anonymous · Answer

that step you type when you first see the question

zarkon · Answer

that's what I usually do

anonymous · Answer

how do u see it

zarkon · Answer

when I see a function of x raised to a function of x I always rewrite it in terms of exponential and log functions

zarkon · Answer

that is...if i plan to differentiate