Finding derivatives of y = x^(a^x), (x+p)^(-1), and rational/trigonometric functions

[mousesgr]

find

1 .derivaive of y, y = x^a^x

2. nth derivaive of (x+p)^-1 p is constant

3. nth derivaive of (ax+b)/(cx+d)

4. nth derivaive of y = sin (^2) x

 

[HallsofIvy]

Looks like homework to me. Is it?

The first one like you need to work with ln(y)= (a^x)ln x.

The others, just calculate two or three derivatives and see if you can spot a pattern.

 

[mousesgr]

i know how to do qs 1

 

[HallsofIvy]

Did you read what I wrote before?

For $$y= x^{(a^x)}$$, take the logarithm of both sides:

ln(y)= a^xln(x). Now differentiate both sides, with respect to x.
(You will need to use the chain rule on the left side and the product rule on the right.)

 

[mousesgr]

Did you read what I wrote before?

For $$y= x^{(a^x)}$$, take the logarithm of both sides:

ln(y)= a^xln(x). Now differentiate both sides, with respect to x.
(You will need to use the chain rule on the left side and the product rule on the right.)

i mean i know how to do the first one now

 

[VietDao29]

For #2, you can use: (a^b)' = b a^{b - 1}.
Then note that (x + p)' = 1.
For example: $$y' = \left( \frac{1}{x + p} \right)' = -\frac{1}{(x + p) ^ 2}$$
$$y'' = (y')' = -\left( \frac{1}{(x + p) ^ 2} \right)' = 2\frac{1}{(x + p) ^ 3}$$
So $$y ^ {(n)} \ = \ ?$$
-----------------------
For #3, you need to arrange $$\frac{ax + b}{cx + d}$$ into something like: $$C + \frac{A}{cx + d}$$, where C, and A = const.
Then you just do the same like #2.
-----------------------
For #4, you can try to take 1st, 2nd, 3rd, 4th, ... derivative of the function and see the rule.
Note that 2sin(x)cos(x) = sin(2x).
Viet Dao,

 

[r3dxP]

find

1 .derivaive of y, y = x^a^x

2. nth derivaive of (x+p)^-1 p is constant

3. nth derivaive of (ax+b)/(cx+d)

4. nth derivaive of y = sin (^2) x

1. ln y = a^x lnx
y'/y = [xa^(x-1)]lnx + [(1/x)a^x]
y' = y[[xa^(x-1)]lnx + [(1/x)a^x]]

2. -1(x+p)^-2
too lazyt to do the rest..