Question

Find the derivative of:f(x)=ln (x^2sin(x)/arctan(x))

Answer 1

Write the function

`f(x)\text{ = }\ln\left((x^2\sin\left(x\right))/(\arctan\left(x\right))\right)`

Next

Apply the chain rule

`\begin{gathered} Let\text{ u =}\left((x^2\sin\left(x\right))/(\arctan\left(x\right))\right) \\ \mathrm{Apply\:the\:Quotient\:Rule}:\quad \left((f)/(g)\right)^'=(f\:'\cdot g-g'\cdot f)/(g^2) \\ =((d)/(dx)\left(x^2\sin \left(x\right)\right)\arctan \left(x\right)-(d)/(dx)\left(\arctan \left(x\right)\right)x^2\sin \left(x\right))/(\left(\arctan \left(x\right)\right)^2) \\ (d)/(dx)\left(x^2\sin\left(x\right)\right)=(d)/(dx)\left(x^2\sin\left(x\right)\right) \\ \\ (d)/(dx)\left(\arctan\left(x\right)\right)\text{ = }(1)/(x^2+1) \\ \\ =(\left(2x\sin \left(x\right)+\cos \left(x\right)x^2\right)\arctan \left(x\right)-(1)/(x^2+1)x^2\sin \left(x\right))/(\left(\arctan \left(x\right)\right)^2) \\ \\ Simplify \\ \\ (du)/(df)=(\arctan\left(x\right)\left(2x\sin\left(x\right)+x^2\cos\left(x\right)\right)\left(x^2+1\right)-x^2\sin\left(x\right))/(\arctan^2\left(x\right)\left(x^2+1\right)) \end{gathered}`

Next

`\begin{gathered} f(x)\text{ = In\lparen u\rparen} \\ (df)/(du)\text{ = }(1)/(u) \\ \\ From\text{ chain rule} \\ \\ f^(\prime)(x)=\text{ }(du)/(dx)\text{ }*\text{ }(df)/(du) \\ \\ =(1)/((x^2\sin\left(x\right))/(\arctan\left(x\right)))(d)/(dx)\left((x^2\sin\left(x\right))/(\arctan\left(x\right))\right) \\ \\ =(1)/((x^2\sin \left(x\right))/(\arctan \left(x\right)))\cdot (\arctan \left(x\right)\left(2x\sin \left(x\right)+x^2\cos \left(x\right)\right)\left(x^2+1\right)-x^2\sin \left(x\right))/(\arctan ^2\left(x\right)\left(x^2+1\right)) \\ \\ =(\arctan \left(x\right)\left(x^2\cos \left(x\right)+2x\sin \left(x\right)\right)\left(x^2+1\right)-x^2\sin \left(x\right))/(x^2\sin \left(x\right)\arctan \left(x\right)\left(x^2+1\right)) \\ \\ \end{gathered}`

Final answer

The derivative of the function is given below:

`f^(\prime)(x)\text{ }=(\arctan\left(x\right)\left(x^2\cos\left(x\right)+2x\sin\left(x\right)\right)\left(x^2+1\right)-x^2\sin\left(x\right))/(x^2\sin\left(x\right)\arctan\left(x\right)\left(x^2+1\right))`