Question

Find the derivative in the form dy/dx

Answer 1

ANSWER

$\frac{dy}{dx} = x{(3x)}^{ {x}^{2} } (ln(9 {x}^{2} ) + 1)$

EXPLANATION

The given equation is;

$y = {(3x)}^{ {x}^{2} }$

Take natural log of both sides:

$ln(y) = ln({(3x)}^{ {x}^{2} } )$

This implies that,

$ln(y) = {x}^{2} ln({(3x)})$

We differentiate to get,

$\frac{1}{y} \frac{dy}{dx} = ln(3x) (2x) + {x}^{2} ( \frac{1}{x} )$

This simplifies to

$\frac{dy}{dx} =y (2x ln(3x) + x)$We substitute for y to get:

$\frac{dy}{dx} = {(3x)}^{ {x}^{2} } (2x ln(3x) + x)$

Or

$\frac{dy}{dx} = x{(3x)}^{ {x}^{2} } (2ln(3x) + 1)$

$\frac{dy}{dx} = x{(3x)}^{ {x}^{2} } (ln(9 {x}^{2} ) + 1)$