Differentiation Help: f(x) = x^e + 1/(x sqrt[10])

[dblu]

Hi!

I have a problem to find the first derivative:

. . . . .\(\displaystyle f(x)\, =\, x^e\, +\, \dfrac{1}{x^{ \sqrt{\strut 10\,}}}\)

I know that via power rule the first term becomes:

. . . . .$\displaystyle e\, x^{e-1}$

but the second term has me stumped. The book says it is:

. . . . .\(\displaystyle -\left(\, \dfrac{\sqrt{\strut 10\,}}{x^{\left(1\, +\, \sqrt{\strut 10\,}\right)}}\,\right)\)

and I just can't wrap my head around how this is happening. I've tried changing the second term into negative exponents (i.e. (x*10^1/2)^-1) and I still can't seem to unravel the sorcery and end up with anything remotely close. Please help!

 

[Deleted member 4993]

Hi!

I have a problem to find the first derivative:

. . . . .\(\displaystyle f(x)\, =\, x^e\, +\, \dfrac{1}{x^{ \sqrt{\strut 10\,}}}\)

I know that via power rule the first term becomes:

. . . . .$\displaystyle e\, x^{e-1}$

but the second term has me stumped. The book says it is:

. . . . .\(\displaystyle -\left(\, \dfrac{\sqrt{\strut 10\,}}{x^{\left(1\, +\, \sqrt{\strut 10\,}\right)}}\,\right)\)

and I just can't wrap my head around how this is happening. I've tried changing the second term into negative exponents (i.e. (x*10^1/2)^-1) and I still can't seem to unravel the sorcery and end up with anything remotely close. Please help!

Looking at the book's answer I surmise that your function to be differentiated is:

$\displaystyle \displaystyle{x^e + \dfrac{1}{x^{(\sqrt{10})}}}$

Then the second term is:

$\displaystyle \displaystyle{ \dfrac{1}{x^{(\sqrt{10})}}} \ = \ x^{(-\sqrt{10})}$

Now substitute:

n = -√(10)

and differentiate using the same power rule that you used for the first term and substitute back!

 

[dblu]

Thank you! You are correct, I misread the original problem and was able to work it through with your help. The print in the book is small and hard to decipher, which can make late night calculus a perilous and frustrating endeavor sometimes.

Also, thank you stapel for editing the post to put the problem in text format. I've no idea how to do it but it's much easier to read.