Question

Question #bda7d

Answer 1

`=> (dy)/(dx) = -8x^(-2x) ( ln(x)+1)`

Explanation:

`y = x^(-2x)`

Taking natural logarithms...

`lny = ln x^(-2x)`

Using the law of:

`color(red)(log_gamma alpha^beta -= beta log_gamma alpha`

`=> lny = -2xln(x)`

Now applying implicit differnetiation....

`=> 1/y * (dy)/(dx) = -2ln(x) -2x*(1/x)`

`=> 1/y * (dy)/(dx) = -2ln(x) - 2`

`=> (dy)/(dx) = -2y ( ln(x) + 1 )`

`=> d/(dx) ( x^(-2x) )= -2x^(-2x) ( ln(x)+1)`

`color(blue)(=> d/(dx) ( 4x^(-2x) )= -8x^(-2x) ( ln(x)+1)`

`y = 4x^(-2x)`:

`(dy)/(dx) = -8x^(-2x) ( ln(x) +1):`