Question

How do you find the derivative of `y = (1/3)^(x^2)`?

Answer 1

`dy/dx = -(2xln3)/(3^(x^2))`

Explanation:

`y= (1/3)^(x^2)`

Take the natural logarithm of both sides.

`lny = ln(1/3)^(x^2)`

Use laws of logarithms to simplify.

`lny = x^2ln(1/3)`

Differentiate using implicit differentiation and the product rule.

`1/y(dy/dx) = 2x(ln(1/3)) + x^2(0)`

`dy/dx= (2xln(1/3))/(1/y)`

`dy/dx = (1/3)^(x^2)2xln(1/3)`

`dy/dx = -2x(1/3)^(x^2)ln3 -> "since "ln(1/3) = ln(3^-1) = -ln3`

`dy/dx = -(2xln3)/(3^(x^2))`

Hopefully this helps!