Can someone help with logarithmic differentiation?

Question

i have two problems that i really don't know how to solve, i am getting so frustrated. can anyone help me please :(

differentiate

qs1. d/dx 3^x^(x+2)

qs2. d/dx sqrt((x^2+1)/(x^4+1))

Answer 1

qs 1.

Let y = 3^x^(x+2)

=> ln y = x^(x+2) ln 3

=> ln ln y = (x+2) ln x + ln ln 3

=> 1/(y ln y) * dy/dx = ln x + [(x+2)/x]

=> dy/dx

= y ln y [ ln x + [(x+2)/x] ]

= [3^x^(x+2)] * [x^(x+2) ln 3] * [ ln x + [(x+2)/x] ]

qs2.

Let y = √((x^2+1)/(x^4+1))

=> ln y = (1/2) [ ln(x^2 + 1) - ln(x^4 + 1) ]

=> 1/y dy/dx = (1/2) [ 2x/(x^2+1) - 4x^3/(x^4+1)]

=> dy/dx = y * √((x^2+1)/(x^4+1))

=> dy/dx = √((x^2+1)/(x^4+1)) * [√[(x^2+1)/(x^4+1)]]

Answer 2

The second question doesn't have to be solved logarithmically at all, even though it can be.

If you rearrange sqrt[(x^2+1)/(x^4+1)], you can obtain

[(x^2+1)^(1/2)][(x^4+1)^(-1/2)], which can be differentiated by combined use of the product and chain rules.

Of course, if you're asked specifically to do it logarithmically, then you have to do it logarithmically, but if not, you might as well use this little shortcut to an easier method (Y)