Finding derivatives?

Question

I just need to check my answers. I'm pretty sure I have them all figured out, but I need to see please.

a) y= x^3*2^x

b) y= ln cubed root(6x+7)

c) y= e^ex ln x

d) y= log_4(x+e^x+e)

e)y= sec(1/sqrt x)

Answer 1

I'm in a bit of a rush, but I'll try the first one:

y = x^3 * 2^x

dy/dx = (x^3 * 2^x ln 2) + 2^x * 3x^2

( I presume you don't need the value of ln 2. If you do, it's 0.69315)

Now I'll try the next one:

y = ln(6x + 7)^1/3

dy/dx = 1/[ln(6x + 7)^1/3] * [1/3(6x + 7)^-2/3] * 6

= 1/[ln(6x + 7)^1/3] *[2(6x + 7)^-2/3]

= [2(6x + 7)^-2/3]/[ln(6x + 7)^1/3]

Phew!

Answer 2

The second one can be done more easily by first simplifying the function using properties of logs

y= ln (6x+7)^(1/3) = (1/3) ln (6x+7)

Now the differentiation is trivial.



(In the previous post, the denominator of the expression after "dy/dx=" should not have "ln" in it. )