# Thread: derivative of a function

1. ## derivative of a function

Please, what woudl be the derivative of a function?
I foudn sth like:

$(f(x)=x^a$
$(f(x)'=x^a(a)'$
is that ok???

2. ## Re: derivative of a function

Originally Posted by Boo
Please, what woudl be the derivative of a function?
I foudn sth like:
$(f(x)=x^a$ find $(f(x)'=x^a(a)'$

If $a$ is a non-zero constant then

$\left(x^a\right)^{\prime}=a\left(x\right)^{a-1}$

3. ## Re: derivative of a function

Hello!
I ment, if a = function!
some tol dme it is different!!!

4. ## Re: derivative of a function

Yep, that is different.

$(x^{a(x)})'$

$= (e^{a(x) \ln x})'$

$= e^{a(x) \ln x} \cdot (a'(x) \ln x + a(x) \cdot \frac 1 x)$

$= x^{a(x)} \cdot (a'(x) \ln x + a(x) \cdot \frac 1 x)$

$= x^{a(x)} a'(x) \ln x + a(x) x^{a(x) - 1}$

5. ## Re: derivative of a function

Yes, that is exactly what i ment!!
Do we "add" lnx in that formula or is that a typo? Many thans! I ment, is the original
$x^a$or $x^{alnx}$

6. ## Re: derivative of a function

Originally Posted by Boo
Please, what woudl be the derivative of a function?
I foudn sth like:

$(f(x)=x^a$
$(f(x)'=x^a(a)'$
is that ok???
Logarithmic differentiation works well here...

\displaystyle \begin{align*} y &= x^{a} \\ \ln{(y)} &= \ln{\left( x^a \right)} \\ \ln{(y)} &= a\ln{(x)} \\ \frac{d}{dx} \left[ \ln{(y)} \right] &= \frac{d}{dx} \left[ a\ln{(x)} \right] \\ \frac{d}{dy}\left[ \ln{(y)} \right] \frac{dy}{dx} &= a\,\frac{d}{dx} \left[ \ln{(x)} \right] + \ln{(x)} \,\frac{d}{dx} \left( a \right) \\ \frac{1}{y} \, \frac{dy}{dx} &= a \left( \frac{1}{x} \right) + \ln{(x)}\,\frac{da}{dx} \\ \frac{dy}{dx} &= y \left[ \frac{a}{x} + \ln{(x)} \,\frac{da}{dx} \right] \\ \frac{dy}{dx} &= x^a \left[ \frac{a}{x} + \ln{(x)} \, \frac{da}{dx} \right] \end{align*}

7. ## Re: derivative of a function

Originally Posted by Boo
Yes, that is exactly what i ment!!
Do we "add" lnx in that formula or is that a typo? Many thans! I ment, is the original
$x^a$or $x^{alnx}$
There is no typo. lnx is in the formula, but it is not added but multiplied.
I guess I could also write it as:

$(x^a)' = x^a a' \ln x + ax^{a-1}$

8. ## Re: derivative of a function

got it!!!
many many thanks!!!

9. ## Re: derivative of a function

Originally Posted by Boo
Yes, I see!
How did U get the first line, sorry, I really forgot...
$(x^a(x))'=(e^{a(x)lnx}$
Many thanks¨!!!
It follows from the calculation rules for logarithm and exponentials.

Suppose $q=\ln p$, then:
$p = e^q$ by definition.
If we substitute q, we get:
[1] $p = e^{\ln p}$

We also have the rule:
[2] $\ln p^q = q \ln p$

So to make that first step more explicit.
Using [1] we have:
$x^a = e^{\ln(x^a)}$
And using [2] we have:

$e^{\ln(x^a)} = e^{a \ln x}$