expeditiupc

2021-12-31

What is the derivative of $y=\mathrm{arctan}\left(x\right)$ ?

Mary Nicholson

Beginner2022-01-01Added 38 answers

The derivative of $y=\mathrm{arctan}x$ is $y}^{\prime}=\frac{1}{1+{x}^{2}$

We can derive this by using implicit differentiation.

Since inverse tangent is hard to deal with, we rewrite it as$\mathrm{tan}\left(y\right)=x$

By implicitly differentiating with respect to x,${\mathrm{sec}}^{2}\left(y\right)\cdot {y}^{\prime}=1$

By solbing for y' and using${\mathrm{sec}}^{2}\left(y\right)=1+{\mathrm{tan}}^{2}\left(y\right)$

$y}^{\prime}=\frac{1}{{\mathrm{sec}}^{2}\left(y\right)}=\frac{1}{{\mathrm{tan}}^{2}\left(y\right)$

Hence,$y}^{\prime}=\frac{1}{1+{x}^{2}$

We can derive this by using implicit differentiation.

Since inverse tangent is hard to deal with, we rewrite it as

By implicitly differentiating with respect to x,

By solbing for y' and using

Hence,

alkaholikd9

Beginner2022-01-02Added 37 answers

Step 1: Rearrange $y=\mathrm{arctan}\left(x\right)$ as $\mathrm{tan}\left(y\right)=x$ .

Step 2: Use implicit differentiation to differentiate this with respect to x, which gives us:

$\left(\frac{dy}{dx}\right)\cdot {\left(\mathrm{sec}\left(y\right)\right)}^{2}=1$

Step 3: Rearrange this equation to give us:

$\frac{dy}{dx}=\frac{1}{{\mathrm{sec}}^{2}\left(y\right)}$

Step 2: Use implicit differentiation to differentiate this with respect to x, which gives us:

Step 3: Rearrange this equation to give us:

nick1337

Expert2022-01-08Added 777 answers

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