Find the derivative of the function?

Question

Find the derivative of the function?

$y = {ln tan}^{2} 3 x$ $y=ln tan^2 3x$

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Answer 1 · 2018-03-08T18:59:32

$\frac{d y}{d x} = 12 csc (6 x)$ $dy/dx=12csc(6x)$

Explanation:

We to find the derivative of

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

Use the chain rule, if $y = f (u)$ $y=f(u)$ and $u = g (x)$ $u=g(x)$ then

$\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}$ $dy/dx=dy/(du)(du)/dx$

Let $y = 2 ln (u) \Rightarrow \frac{d y}{d u} = \frac{2}{u}$ $y=2ln(u)=>(dy)/(du)=2/u$

and $u = tan (3 x) \Rightarrow \frac{d u}{d x} = 3 {sec}^{2} (3 x)$ $u=tan(3x)=>(du)/(dx)=3sec^2(3x)$

Thus

$\frac{d y}{d x} = \frac{6}{u} {sec}^{2} (3 x) = \frac{6}{tan (3 x)} {sec}^{2} (3 x) = 6 csc (3 x) sec (3 x)$ $dy/dx=6/usec^2(3x)=6/tan(3x)sec^2(3x)=6csc(3x)sec(3x)$

Or by applying the identity $csc (2 x) = \frac{1}{2} csc (x) sec (x)$ $csc(2x)=1/2csc(x)sec(x)$

$\frac{d y}{d x} = 12 csc (6 x)$ $dy/dx=12csc(6x)$

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Find the derivative of the function?

$y = {ln tan}^{2} 3 x$ $y=ln tan^2 3x$

1 Answer

Explanation:

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$y = {ln tan}^{2} 3 x$ $y=ln tan^2 3x$