Question

Need help for Calc 1 quiz!!! Implicit differentiation of y=tan^-1(x) / x^2 I did the product rule, but I'm lost in how to simplify it....

Answer 1

https://www.wolframalpha.com/

Answer 2

That will help with calculations like that! :3

Answer 3

Yes, but it doesn't show steps.

Answer 4

y = f.g
\[f = \tan^{-1} x \\ g = x^{-2}\]
apply product rule,
\[y \prime = f \prime g + g \prime f \\ f \prime = \frac{1}{1 + x^2} \\ g \prime = -2.x^{-3}\]

Now can you simplify?

Answer 5

\(\bf y=\cfrac{tan^{-1}(x)}{x^2}
\\ \quad \\
\cfrac{dy}{dx}=\cfrac{dy}{dx}\left[ \cfrac{tan^{-1}(x)}{x^2} \right]\implies \cfrac{dy}{dx}=\cfrac{dy}{dx}[tan^{-1}(x)\cdot x^{-2}]
\\ \quad \\
\left( \cfrac{1}{1+x^2}\cdot x^{-2} \right)+\left( tan^{-1}(x)\cdot -2x^{-3}\right)
\\ \quad \\
\cfrac{1}{x^2(1+x^2)}+tan^{-1}(x)\cdot -2\cdot \cfrac{1}{x^3}\implies
\cfrac{1}{x^2+x^4}-\cfrac{2tan^{-1}(x)}{x^3}\)

Answer 6

Thank you jdoe0001 !! The way you did it was wayyy better to understand how parts fit together. :)

Answer 7

I did good too, that's not fair. :)