(1 point)
Find $d y / d x$ using the method of logarithmic differentiation when $y=\left(5+9 x^{2}\right)^{\ln (x)}$.
\[
d y / d x=
\]

Answer

Expert–verified

Hide Steps

Answer

$\boxed{\frac{dy}{dx} = (5+9x^2)^{\ln(x)}\left(\frac{18x\ln(x)}{5+9x^2} + \frac{\ln(5+9x^2)}{x}\right)}$

Steps

Step 1 ：Let $ y = (5 + 9x^2)^{\ln(x)} $

Step 2 ：Take the natural logarithm of both sides: $ \ln(y) = \ln((5 + 9x^2)^{\ln(x)}) $

Step 3 ：Differentiate both sides with respect to $ x $: $ \frac{dy}{dx} = \frac{18x\ln(x)}{5+9x^2} + \frac{\ln(5+9x^2)}{x} $

Step 4 ：Solve for $ \frac{dy}{dx} $: $ \frac{dy}{dx} = (5+9x^2)^{\ln(x)}\left(\frac{18x\ln(x)}{5+9x^2} + \frac{\ln(5+9x^2)}{x}\right) $

Step 5 ：$\boxed{\frac{dy}{dx} = (5+9x^2)^{\ln(x)}\left(\frac{18x\ln(x)}{5+9x^2} + \frac{\ln(5+9x^2)}{x}\right)}$