Derivative Problems with Exponential and Logarithmic Functions

[TommyLF]

I need a little help on the following problems:

Find the 2nd derivative at the indicated point:
y= x(ln x) At the point (1,0)

I got to:

y''= $$\frac{1}{x}$$

So do I just sub in 1 for x? And then the answer would be 1?

Also, I need help on the following problem:

Evaluate the first derivative at the given value X: x=0

[4^(3x)(x^3 - x + 1)^(1/5)] (And then all of that raised to the -2)
-------------------------
(x^2 + x + 1)^(4)

 

[JasonRox]

I need a little help on the following problems:

Find the 2nd derivative at the indicated point:
y= x(ln x) At the point (1,0)

I got to:

y''= $$\frac{1}{x}$$

So do I just sub in 1 for x? And then the answer would be 1?

Also, I need help on the following problem:

Evaluate the first derivative at the given value X: x=0

[4^(3x)(x^3 - x + 1)^(1/5)] (And then all of that raised to the -2)
-------------------------
(x^2 + x + 1)^(4)

Yeah, basically put 1 in the equation for the first question.

For the second one, I see no trick. I guess you have to painfully use the Quotient Rule, Chain Rule and Product in the right order. Have fun doing that. :S

Actually, I'd probably do it using first principles. It seems like the easier way out. (First principles is finding the derivative using the limit definition.)

 

[HallsofIvy]

JasonRox, you have a wicked sense of humor! Anyone trying to find the derivative of that second function using the limit of the difference quotient is guarenteed to go insane!

TommyLF, I would recommend going ahead and incorporating that -2 exponent into the numerator and, because that is a negative exponent, writing the whole thing as a product of terms with negative exponents:
$$\frac{[4^{3x}(x^3- x+ 1)^{1/5}]^{-2}}{(x^2+ x+ 1)^4}= 4^{-6x}(x^2-x+1)^{-2/5}(x^2+ x+ 1)^{-4}$$
and use the product rule and chain rule.